The SAT provides a small reference sheet at the start of each Math module containing basic geometry formulas (area of a circle, volume of a box, the Pythagorean theorem, and a few others). Students who prepare only to the level of that reference sheet are under-prepared for the Digital SAT. The full set of formulas and concepts tested on the Digital SAT is substantially larger, and most of the tested formulas are NOT on the provided reference sheet.

This article is the complete formula and concept reference for every formula, rule, and relationship tested on the Digital SAT Math section. It is organized by domain (Algebra, Advanced Math, Geometry, Statistics and Probability, and Key Rules) and each entry includes a one-sentence explanation of when you need it. Use this reference during your final review weeks to confirm that every formula is memorized and that you know the specific question context that triggers its use.

For the trigonometry context where the right triangle formulas apply, see the SAT Math right triangles and unit circle guide. For the complete Math section format and question distribution, see the complete Digital SAT guide. For timed practice applying these formulas, the free SAT Math practice questions on ReportMedic provide Digital SAT-format problems across all formula areas.

How to Use This Reference Sheet

This reference sheet is designed for two uses: study and review.

During study (weeks before the exam): work through each domain section and identify the formulas you do not yet have memorized. For each unknown formula, write it on a flash card, study the one-sentence explanation, and practice applying it to one or two sample problems. Prioritize the formulas most heavily tested on the Digital SAT (marked with usage frequency indicators in each section).

During final review (1 to 2 days before the exam): read through the complete reference sheet and confirm that every formula is familiar. For any formula that feels uncertain, spend 5 to 10 minutes working through 2 to 3 practice problems that use it. Do not try to learn new formulas in the 24 hours before the exam; use this reference to confirm what you already know.

The organizing principle of the reference: formulas are grouped by topic domain, not by difficulty. Easy and hard formulas appear together in each domain because the Digital SAT does not separate by difficulty within a domain. Knowing the domain context of each formula (Algebra vs Geometry, for example) helps with rapid recall during the exam.

Domain 1: Algebra Formulas

The Algebra domain accounts for approximately 35 percent of Digital SAT Math questions. Most Algebra formulas are not on the provided SAT reference sheet and must be memorized.

SLOPE FORMULA: Formula: m = (y2 minus y1) / (x2 minus x1) When you need it: any question that gives two points and asks for the slope, the rate of change, or asks whether lines are parallel or perpendicular. Usage: parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other (if one slope is m, the perpendicular slope is minus 1/m).

SLOPE-INTERCEPT FORM: Formula: y = mx + b (m = slope, b = y-intercept) When you need it: writing or identifying a linear equation from a description, finding the y-intercept directly (it is the constant term b), or graphing a line on Desmos. Note: the y-intercept is the value of y when x = 0; substitute x = 0 to find it directly.

POINT-SLOPE FORM: Formula: y minus y1 = m(x minus x1) (where (x1, y1) is a known point and m is the slope) When you need it: writing the equation of a line when given a point and a slope (or two points, from which slope is computed first). Usage: preferred form when one specific point is given rather than the y-intercept.

STANDARD FORM FOR A LINE: Formula: Ax + By = C (A, B, C are integers) When you need it: interpreting a linear equation given in standard form; finding x and y intercepts directly. The x-intercept is C/A (set y = 0); the y-intercept is C/B (set x = 0). Tip: Desmos accepts this form directly without converting to slope-intercept.

QUADRATIC FORMULA: Formula: x = (-b plus or minus sqrt(b squared minus 4ac)) / 2a, where the quadratic is ax squared + bx + c = 0. When you need it: solving any quadratic equation that does not factor easily; any question asking for the exact roots of a quadratic. Critical note: the formula applies when ax squared + bx + c = 0. Rearrange the equation to this standard form before identifying a, b, and c.

VERTEX FORMULA (x-COORDINATE OF VERTEX): Formula: x = minus b / (2a), where the quadratic is ax squared + bx + c. When you need it: finding the vertex x-coordinate of a parabola (and the minimum or maximum value of a quadratic function), or the axis of symmetry. Usage: after finding x = minus b/(2a), substitute back into the original expression to find the y-coordinate of the vertex (the minimum or maximum value).

VERTEX FORM OF A QUADRATIC: Formula: y = a(x minus h) squared + k, where (h, k) is the vertex. When you need it: identifying the vertex from the equation, completing the square (which converts standard form to vertex form), or confirming a transformation. Usage: if a is positive, the parabola opens upward (minimum at (h, k)). If a is negative, it opens downward (maximum at (h, k)).

DISCRIMINANT ANALYSIS: Formula: discriminant = b squared minus 4ac. When you need it: determining how many real solutions a quadratic equation has without solving it. Outcomes: discriminant greater than 0 means two distinct real solutions. Discriminant equal to 0 means exactly one real solution (a repeated root). Discriminant less than 0 means no real solutions (two complex solutions).

COMPLETING THE SQUARE (STANDARD PROCEDURE): Procedure for ax squared + bx + c: Step 1: factor out a from the first two terms: a(x squared + (b/a)x) + c. Step 2: complete the square inside: half of (b/a) is b/(2a); square it to get b squared/(4a squared). Step 3: add and subtract: a(x squared + (b/a)x + b squared/(4a squared)) + c minus a times b squared/(4a squared). Step 4: factor the perfect square: a(x + b/(2a)) squared + c minus b squared/(4a). When you need it: finding the vertex of a parabola, solving quadratics that do not factor, or rewriting a quadratic in vertex form. Note: the vertex x-coordinate from this procedure confirms the vertex formula x = minus b/(2a). The completing-the-square procedure is the derivation behind the formula; both produce the same x-coordinate.

EXPONENT RULES: Product rule: x to the power a times x to the power b = x to the power (a + b). Multiply same base: add exponents. Quotient rule: x to the power a divided by x to the power b = x to the power (a minus b). Divide same base: subtract exponents. Power rule: (x to the power a) to the power b = x to the power (a times b). Power of a power: multiply exponents. Zero exponent: x to the power 0 = 1 (for any nonzero x). Negative exponent: x to the power (minus a) = 1 / (x to the power a). Negative exponent means reciprocal. When you need them: simplifying exponential expressions, solving exponential equations, working with scientific notation.

RADICAL-EXPONENT CONVERSION: Formula: the n-th root of x = x to the power (1/n). The n-th root of x to the power m = x to the power (m/n). When you need it: converting between radical notation and fractional exponents; simplifying expressions containing both radicals and exponents. Example: cube root of x squared = x to the power (2/3).

ABSOLUTE VALUE DEFINITION: Formula: |x| = x if x is non-negative; |x| = minus x if x is negative. Alternative interpretation: |x| is the distance from x to 0 on the number line. |x minus a| is the distance from x to a. When you need it: solving absolute value equations and inequalities, interpreting |x minus a| as a distance condition. The distance interpretation is especially useful for word problems: “all values of x within 5 of 12” translates to |x minus 12| less than or equal to 5, which gives 7 less than or equal to x less than or equal to 17. The verbal “within d of a” always becomes |x minus a| less than or equal to d.

Domain 2: Advanced Math Formulas

The Advanced Math domain accounts for approximately 35 percent of Digital SAT Math questions and is the domain with the highest average difficulty.

FACTORED FORM OF A QUADRATIC: Formula: ax squared + bx + c = a(x minus r1)(x minus r2), where r1 and r2 are the roots. When you need it: identifying zeros of a quadratic from factored form, converting between factored and standard form, writing a quadratic equation from its roots.

DIFFERENCE OF SQUARES: Formula: a squared minus b squared = (a + b)(a minus b). When you need it: factoring expressions of the form x squared minus 9 = (x+3)(x-3), or any difference-of-squares pattern. High-frequency trigger: (x squared minus 4)/(x minus 2) can be simplified by factoring the numerator as (x+2)(x-2), then canceling (x minus 2).

PERFECT SQUARE TRINOMIALS: Formulas: (a + b) squared = a squared + 2ab + b squared; (a minus b) squared = a squared minus 2ab + b squared. When you need them: recognizing perfect square patterns in algebraic expressions, completing the square, converting between expanded and factored forms.

SUM/DIFFERENCE OF CUBES: Formulas: a cubed + b cubed = (a + b)(a squared minus ab + b squared); a cubed minus b cubed = (a minus b)(a squared + ab + b squared). When you need them: factoring cubic expressions of the form x cubed plus or minus 8 (where 8 = 2 cubed). Digital SAT frequency: low to moderate; these appear primarily on harder Advanced Math questions.

POLYNOMIAL REMAINDER THEOREM: Statement: when f(x) is divided by (x minus a), the remainder equals f(a). When you need it: any question asking for the remainder of polynomial division, or asking for which value of a makes (x minus a) a factor (set f(a) = 0 and solve). Speed advantage: evaluating f(a) takes 15 seconds; polynomial long division takes 2 to 3 minutes. Always use the remainder theorem.

FUNCTION NOTATION: f(x): the output of function f when the input is x. f(a) = b means: when x = a is substituted into f, the output is b. f(g(x)): the composition of f and g; evaluate g first, then substitute the result into f. When you need it: any function evaluation or composition question; piecewise function questions (where the correct piece is selected based on the input value).

PIECEWISE FUNCTION EVALUATION: Procedure: identify which condition the input x satisfies, then evaluate the corresponding formula. When you need it: questions involving piecewise-defined functions; compositions where one function is piecewise. Key habit: always check which piece applies before evaluating; the domain condition is part of the function definition.

COMPLEX NUMBER OPERATIONS: Definition: i = sqrt(minus 1), so i squared = minus 1. i-power cycle: i to the 1 = i; i squared = minus 1; i cubed = minus i; i to the 4 = 1; then the pattern repeats. Addition/subtraction: combine real parts and imaginary parts separately: (a + bi) plus (c + di) = (a + c) + (b + d)i. Multiplication: FOIL, then replace i squared with minus 1: (a + bi)(c + di) = ac + adi + bci + bdi squared = (ac minus bd) + (ad + bc)i. Division (conjugate method): multiply numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) times (c minus di)/(c minus di). When you need them: any question involving i, complex solutions of quadratics (discriminant less than 0), or complex number arithmetic.

CONJUGATE OF A COMPLEX NUMBER: Formula: the conjugate of (a + bi) is (a minus bi). Property: (a + bi)(a minus bi) = a squared + b squared (a real number). When you need it: complex number division; simplifying expressions with imaginary denominators.

EXPONENTIAL FUNCTION MODELS: General form: f(t) = a times b to the power t (a = initial value, b = growth/decay factor). For percent growth: b = 1 + r (r is the decimal growth rate). For percent decay: b = 1 minus r. For doubling every d periods: f(t) = a times 2 to the power (t/d). For halving every d periods: f(t) = a times (1/2) to the power (t/d). When you need them: any exponential modeling question; identifying the growth or decay factor from a given percent rate.

LINEAR vs EXPONENTIAL IDENTIFICATION (TWO-TEST): Linear test: constant differences between consecutive y-values (equal x-spacing). Exponential test: constant ratios between consecutive y-values (equal x-spacing). When you need it: any question asking which model (linear or exponential) better fits given data, or asking for the model type from a table of values.

Domain 3: Geometry and Trigonometry Formulas

The Geometry and Trigonometry domain accounts for approximately 15 percent of Digital SAT Math questions. The SAT provides some geometry formulas; the following includes all provided formulas PLUS the additional formulas not on the reference sheet.

PROVIDED BY SAT REFERENCE SHEET: Area of a rectangle: A = lw. Area of a triangle: A = (1/2)bh. Area of a circle: A = pi r squared. Circumference of a circle: C = 2 pi r. Volume of a rectangular prism: V = lwh. Volume of a cylinder: V = pi r squared h. Volume of a sphere: V = (4/3) pi r cubed. Volume of a cone: V = (1/3) pi r squared h. Volume of a pyramid: V = (1/3) lwh. Pythagorean theorem: a squared + b squared = c squared (a and b are legs, c is hypotenuse). Special right triangle 30-60-90: sides are in ratio 1 : sqrt(3) : 2. Special right triangle 45-45-90: sides are in ratio 1 : 1 : sqrt(2). Number of degrees in a circle: 360. Number of radians in a circle: 2 pi. Sum of angles in a triangle: 180 degrees.

NOT ON SAT REFERENCE SHEET (MUST MEMORIZE):

DISTANCE FORMULA: Formula: d = sqrt((x2 minus x1) squared + (y2 minus y1) squared). When you need it: any question asking for the distance between two points in the coordinate plane. Desmos shortcut: plot the two points and use the click-to-show-coordinates feature to confirm the distance formula result.

MIDPOINT FORMULA: Formula: midpoint = ((x1 + x2)/2, (y1 + y2)/2). When you need it: any question asking for the midpoint of a line segment, or any question where the midpoint is given and one endpoint is unknown (solve for the other).

CIRCLE EQUATION (STANDARD FORM): Formula: (x minus h) squared + (y minus k) squared = r squared, where (h, k) is the center and r is the radius. When you need it: finding the center and radius from a circle equation, writing the equation from given center and radius, or working with circle-line intersection problems. Note: the general form ax squared + by squared + cx + dy + e = 0 must be converted to standard form by completing the square in both x and y.

ARC LENGTH: Formula: arc length = (central angle / 360) times circumference = (theta / 360) times 2 pi r. In radians: arc length = r times theta. When you need it: any question asking for the length of an arc of a circle given the central angle and radius.

SECTOR AREA: Formula: sector area = (central angle / 360) times pi r squared. In radians: sector area = (1/2) r squared times theta. When you need it: any question asking for the area of a “pie slice” portion of a circle given the central angle and radius.

ANGLE PAIR RELATIONSHIPS: Supplementary angles: two angles that sum to 180 degrees. Complementary angles: two angles that sum to 90 degrees. Vertical angles: angles formed by two intersecting lines opposite each other. Vertical angles are equal. Linear pair: two adjacent angles that form a straight line. They are supplementary (sum = 180 degrees). When you need them: any multi-angle problem; the angle-sum relationships chain through a sequence of steps to find unknown angles.

PARALLEL LINES AND TRANSVERSAL: When a transversal crosses two parallel lines: Corresponding angles: equal (in the same position at each intersection). Alternate interior angles: equal (on opposite sides of the transversal between the parallel lines). Same-side interior angles (co-interior): supplementary (sum to 180 degrees). Alternate exterior angles: equal. When you need them: any question involving parallel lines and a transversal; “find the angle” questions that require identifying corresponding or alternate angles.

EXTERIOR ANGLE THEOREM: Formula: the exterior angle of a triangle equals the sum of the two non-adjacent interior angles. When you need it: any question involving an exterior angle of a triangle and the interior angles. Example: if exterior angle = 120 degrees and one non-adjacent interior angle = 70 degrees, the other non-adjacent interior angle = 50 degrees.

POLYGON INTERIOR ANGLE SUM: Formula: sum of interior angles of an n-sided polygon = (n minus 2) times 180 degrees. For a regular polygon, each interior angle = (n minus 2) times 180 / n degrees. When you need it: any question asking for the sum of interior angles of a polygon, or the measure of each interior angle of a regular polygon. The formula derives from dividing any polygon into (n minus 2) non-overlapping triangles by drawing diagonals from one vertex; each triangle contributes 180 degrees. For common polygons: triangle (n=3) has sum 180 degrees; quadrilateral (n=4) has 360 degrees; pentagon (n=5) has 540 degrees; hexagon (n=6) has 720 degrees.

SIMILAR TRIANGLES: Property: corresponding sides of similar triangles are proportional; corresponding angles are equal. Ratios: if triangles ABC and DEF are similar with ratio k, then all lengths in DEF are k times the corresponding lengths in ABC. Areas scale as k squared. When you need it: any question involving similar triangles, shadow problems, or scale factor problems. Similar triangles arise in many non-obvious contexts: whenever a line parallel to one side of a triangle is drawn, it creates a smaller similar triangle; whenever two lines from the same point intersect two parallel lines, similar triangles are formed. Recognizing the similar-triangle structure within a complex figure is the primary skill tested in similar-triangle questions.

PYTHAGOREAN TRIPLES (common): 3-4-5 and multiples (6-8-10, 9-12-15). 5-12-13. 8-15-17. 7-24-25. When you need them: recognizing right triangles from side lengths without using the full a squared + b squared = c squared computation.

COORDINATE GEOMETRY: LINE THROUGH TWO POINTS: Procedure: find slope m = (y2 minus y1)/(x2 minus x1), then use point-slope form y minus y1 = m(x minus x1), simplify. When you need it: writing the equation of a line when two points are given; finding the y-intercept from two points.

TRIGONOMETRIC RATIOS (SOH-CAH-TOA): In a right triangle with acute angle theta: sin(theta) = opposite / hypotenuse. cos(theta) = adjacent / hypotenuse. tan(theta) = opposite / adjacent = sin(theta) / cos(theta). When you need them: any question involving the length of sides in a right triangle when an angle measure is given, or finding an angle from side lengths.

COMPLEMENTARY ANGLE TRIG IDENTITIES: sin(theta) = cos(90 minus theta). cos(theta) = sin(90 minus theta). When you need them: questions that provide sin(x) = some value and ask for cos(90 minus x), or any combination of complementary angles in a right triangle. Derivation: in a right triangle, the two acute angles sum to 90 degrees, so sin of one acute angle = cos of the other. Digital SAT application: these identities are tested in a specific high-frequency question format: “In a right triangle, angle A and angle B are the two acute angles. If sin(A) = 4/5, what is cos(B)?” Since A + B = 90, cos(B) = cos(90 minus A) = sin(A) = 4/5. The answer is immediate once the identity is recalled.

UNIT CIRCLE KEY VALUES: sin(0) = 0, cos(0) = 1. sin(30) = 1/2, cos(30) = sqrt(3)/2. sin(45) = sqrt(2)/2, cos(45) = sqrt(2)/2. sin(60) = sqrt(3)/2, cos(60) = 1/2. sin(90) = 1, cos(90) = 0. When you need them: exact trigonometric value questions; verifying solutions to trigonometric equations. Memory aid for sin values at 0, 30, 45, 60, 90 degrees: sqrt(0)/2 = 0, sqrt(1)/2 = 1/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2 = 1. The pattern sqrt(k)/2 for k = 0, 1, 2, 3, 4 gives all five sin values in order. Cos values are the reverse sequence.

RADIAN-DEGREE CONVERSION: Formula: radians = degrees times pi/180. Degrees = radians times 180/pi. Key conversions: pi/6 = 30 degrees, pi/4 = 45 degrees, pi/3 = 60 degrees, pi/2 = 90 degrees, pi = 180 degrees, 2 pi = 360 degrees. When you need it: any question mixing radians and degrees, or requiring radian-to-degree conversion for trig function evaluation.

SURFACE AREA FORMULAS (not provided by SAT): Rectangular prism: SA = 2(lw + lh + wh). Cylinder: SA = 2 pi r squared + 2 pi r h. Sphere: SA = 4 pi r squared. When you need them: questions explicitly asking for surface area rather than volume.

3D SCALING PRINCIPLE: When all linear dimensions of a solid are multiplied by k: volume scales by k cubed; surface area scales by k squared. When only one dimension changes (e.g., height of a cylinder): multiply the formula by k raised to the power of that dimension’s occurrence in the formula. When you need it: “if the radius is doubled, what happens to the volume?” type questions.

Domain 4: Statistics and Probability Formulas

The Problem Solving and Data Analysis domain accounts for approximately 15 percent of Digital SAT Math questions.

MEAN (ARITHMETIC AVERAGE): Formula: mean = (sum of all values) / (number of values). When you need it: any question asking for the mean; finding a missing value when the mean is given; working backward from mean to sum. Working backward: if mean = m and there are n values, sum = m times n.

WEIGHTED MEAN: Formula: weighted mean = (sum of (value times weight)) / (sum of weights). When you need it: questions involving grade averages with different credit weights, or population averages for two groups.

MEDIAN: Definition: the middle value when data is sorted in order. For an even number of values, the median is the average of the two middle values. When you need it: any question asking for the median; questions about the effect of adding or removing a value on the median vs mean.

MODE: Definition: the value that appears most frequently in a data set. When you need it: questions explicitly asking for the mode; bimodal distributions (where two values tie for most frequent).

RANGE: Formula: range = maximum value minus minimum value. When you need it: any question asking for the spread of a data set by range.

STANDARD DEVIATION INTERPRETATION: Definition: a measure of how spread out values are around the mean. Higher standard deviation = more spread. Important property: adding a constant to every value shifts the mean but does not change the standard deviation. Multiplying every value by a constant multiplies both mean and standard deviation by that constant. When you need it: questions about spread, comparing variability between two data sets, or answering “what happens to the standard deviation when…” questions.

PROBABILITY FORMULA: Formula: P(event) = (number of favorable outcomes) / (total number of equally likely outcomes). When you need it: any basic probability question; interpreting probabilities from tables or proportions.

COMPLEMENTARY PROBABILITY: Formula: P(not A) = 1 minus P(A). When you need it: any question asking for the probability that an event does NOT occur; questions where the complement is easier to compute.

CONDITIONAL PROBABILITY: Formula: P(A given B) = P(A and B) / P(B) = (count of A and B) / (count of B). When you need it: any question involving “given that” language; two-way table probability questions where the denominator is a row or column total, not the grand total.

INDEPENDENT EVENTS: Definition: events A and B are independent if P(A and B) = P(A) times P(B). When you need it: probability questions involving two sequential events that do not affect each other (like flipping a coin twice). The Digital SAT tests independence in two ways: directly (asking whether two events are independent given their probabilities) and computationally (finding P(A and B) when independence is stated). For the computational version: identify P(A) and P(B) and multiply them directly. No conditional probability adjustment is needed because independence means P(A given B) = P(A).

COUNTING PRINCIPLE: Formula: if event 1 can occur in m ways and event 2 can occur in n ways, the number of ways both can occur in sequence is m times n. When you need it: counting problems involving ordered selections, arrangements, or sequences of choices.

COMBINATIONS FORMULA: Formula: C(n, k) = n! / (k! times (n minus k)!), where n is the total and k is the number chosen. C(n, k) counts the number of ways to choose k items from n items without regard to order. When you need it: questions asking “how many ways can a group of k be chosen from n people?” where order does not matter.

PERMUTATIONS FORMULA: Formula: P(n, k) = n! / (n minus k)!, where n is the total and k is the number chosen. P(n, k) counts ordered arrangements of k items from n items. When you need it: questions asking “how many ways can k items be arranged from n?” where order matters (passwords, rankings, etc.).

LINE OF BEST FIT (REGRESSION LINE) INTERPRETATION: Slope: the average change in y for each one-unit increase in x. Y-intercept: the predicted value of y when x = 0 (valid only if x = 0 is within the data range). R-squared: closer to 1.0 means a better fit; closer to 0 means a poor fit. When you need it: any question asking what the slope or y-intercept represents in the context of a regression line.

PERCENTILE: Definition: the k-th percentile is the value below which k percent of the data falls. When you need it: questions about how a given value compares to a distribution; interpreting percentile rankings.

SAMPLING AND INFERENCE: Random sample: a sample where every member of the population has an equal chance of being selected. Margin of error: the range within which the true population parameter is estimated to fall. When you need them: questions about the validity of conclusions from sample data; questions about what inferences can and cannot be drawn from a given sample.

Domain 5: Key Rules and Identities

The following rules apply across multiple domains and are among the most frequently tested concepts on the Digital SAT Math section.

INEQUALITY FLIP RULE: Rule: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign flips. Example: if minus 2x less than 6, divide both sides by minus 2 and flip: x greater than minus 3. When you need it: any inequality solution that requires multiplying or dividing by a negative number. Prevention habit (from Article 23): write “FLIP” before performing the operation to avoid forgetting.

ABSOLUTE VALUE INEQUALITY CASES: For |expression| less than k: the solution is minus k less than expression less than k (a bounded interval). For |expression| greater than k: the solution is expression less than minus k OR expression greater than k (two separate intervals). When you need it: solving absolute value inequalities; interpreting “within k units of a” as |x minus a| less than or equal to k.

PROPORTIONALITY AND DIRECT VARIATION: Direct variation: y = kx (y is directly proportional to x, k is the constant of proportionality). Inverse variation: y = k/x (y is inversely proportional to x). When you need them: questions about proportional relationships; percent problems where a quantity scales proportionally. For direct variation: if y is proportional to x and one (x, y) pair is known, find k = y/x, then use y = kx for any other x-value. For inverse variation: if y is inversely proportional to x and one pair is known, find k = xy, then use y = k/x for any other value. The key property of inverse variation: as x increases, y decreases, and the product xy is always constant.

PERCENT CHANGE FORMULA: Formula: percent change = (new minus old) / old times 100 percent. When you need it: any question asking for the percent increase or percent decrease between two values. Prevention habit: always label “old” and “new” before computing to avoid using the wrong denominator.

PERCENT OF A NUMBER: Formula: percent of a number = (percent / 100) times the number, or decimal times number. When you need it: direct “what is P percent of N?” questions; working backward (“N is P percent of what number?”).

SUCCESSIVE PERCENT CHANGES: Formula: multiply the multipliers. For increase by r1 then decrease by r2: final = original times (1 + r1) times (1 minus r2). When you need it: any question involving two or more sequential percent changes; always multiply multipliers, never add rates.

SYSTEMS OF EQUATIONS: For two linear equations with two unknowns: the solution is the intersection point of the two lines. Substitution method: solve one equation for one variable, substitute into the other. Elimination method: add or subtract equations to eliminate one variable. Desmos method: graph both equations, click intersection (fastest for ordered numerical answer choices). When you need it: any two-variable system of equations; also systems involving quadratics (one equation is quadratic).

SYSTEMS WITH SPECIAL SOLUTIONS: No solution: parallel lines (same slope, different y-intercept). Algebraically: contradiction like 5 = 7. Infinite solutions: identical lines (same slope, same y-intercept). Algebraically: identity like 0 = 0. When you need it: questions asking for the value of a parameter that makes a system have no solution or infinite solutions (covered in detail in Article 22 Type 1).

RATE-WORK-TIME FORMULA: Formula: rate = work / time, or work = rate times time. For combined work: combined rate = sum of individual rates. 1/a + 1/b = 1/time to complete together. When you need it: any “how long does it take two workers together?” or “at what rate is the tank being filled?” question.

DISTANCE-RATE-TIME FORMULA: Formula: distance = rate times time (d = rt). When you need it: any problem involving a moving object; problems with two objects moving toward or away from each other (set up separate d = rt equations and combine). For two objects moving toward each other: the time until they meet is the total distance divided by the sum of their speeds. For two objects moving in the same direction: the time until the faster one catches the slower one is the initial separation divided by the difference of their speeds. Both setups follow from writing d = rt for each object and equating total distances.

MIXTURE PROBLEMS: Setup: (percent of substance A) times (amount of A) + (percent of substance B) times (amount of B) = (percent of mixture) times (total amount). When you need it: any question about mixing two solutions or substances with different concentrations.

I-POWER CYCLE: i to the 1 = i. i squared = minus 1. i cubed = minus i. i to the 4 = 1. Pattern repeats every 4 powers. To find i to the n: divide n by 4 and use the remainder: remainder 1 = i, remainder 2 = minus 1, remainder 3 = minus i, remainder 0 = 1. When you need it: any question asking for a high power of i (like i to the 47); powers of i in complex number expressions.

TRIANGLE INEQUALITY THEOREM: Rule: the sum of any two sides of a triangle must be greater than the third side. When you need it: questions asking which values could be the length of the third side of a triangle, or whether a given set of lengths can form a triangle. For “what values of x could be the third side of a triangle with sides 7 and 12?”: the third side must satisfy both 7 + x > 12 (x > 5) AND 7 + 12 > x (x < 19). Combined: 5 < x < 19. This double-inequality setup is the standard application of the triangle inequality.

SUM OF EXTERIOR ANGLES: Rule: the sum of the exterior angles of any convex polygon is always 360 degrees. When you need it: questions involving exterior angles of polygons; finding an unknown exterior angle from the others.

Using the SAT-Provided Reference Sheet Effectively

The SAT provides a small reference sheet at the beginning of each Math module. It contains the following formulas: Area formulas for rectangles, triangles, and circles. Circumference formula for circles. Volume formulas for rectangular prisms, cylinders, spheres, cones, and pyramids. The Pythagorean theorem. Properties of special right triangles (30-60-90 and 45-45-90). The number of degrees in a circle (360) and a triangle (180). The number of radians in a circle (2 pi).

These formulas are provided because the SAT test designers consider them tools rather than knowledge to be memorized. All other formulas in this article are NOT provided and must be memorized.

The reference sheet is available throughout both Math modules. It appears as a button or link at the top of the Bluebook interface. Clicking it opens the reference sheet; closing it returns you to the question. The time cost of opening the reference sheet is approximately 5 to 10 seconds per visit.

Strategic use of the reference sheet: for the provided formulas (area of a circle, volume formulas, etc.), use the reference sheet during the exam if uncertain. Memorizing these formulas is still faster (saves 5 to 10 seconds per use), but using the reference sheet is a reliable fallback. For all other formulas in this article (slope formula, discriminant, percent change, etc.), no reference is provided and they must be recalled from memory.

Formulas That Most Commonly Cause Errors

The following formulas have the highest rate of application errors on the Digital SAT, based on student performance data. Pay special attention to these during final review.

HIGH ERROR RATE FORMULAS:

Vertex formula (x = minus b / 2a): students frequently compute b/2a without the negative sign, producing the wrong x-coordinate. Habit: write the negative sign first before substituting any values.

Percent change denominator: students frequently divide by the new value instead of the old value. Habit: always write “old” and “new” labels before computing.

Conditional probability denominator: students use the total table count instead of the conditional group total as the denominator. Habit: identify the “given” condition and use that group’s total as the denominator.

Discriminant formula: students frequently compute b squared minus 4ac incorrectly when a is negative (double negative errors). Habit: substitute a, b, c values with parentheses: (minus 3) squared minus 4(minus 2)(5).

Arc length formula: students forget to multiply by the fraction of the full circle (theta/360). Habit: write the formula as “fraction of circle times full circumference” to enforce the two-step structure.

Slope between two points: students flip the x-values and y-values (compute x differences in numerator and y differences in denominator). Habit: write “y goes on top” or use the mnemonic “rise over run” (rise = vertical = y change; run = horizontal = x change).

Formula Memorization Strategies

Memorizing the full set of formulas in this reference sheet is achievable with the right strategy. The following three-stage approach works for most students.

Stage one: flash card creation (week one). Create one flash card per formula with the formula on one side and the one-sentence explanation of when to use it on the other. Work through the flash card deck daily for 10 minutes.

Stage two: application practice (weeks two to three). For each formula, work through 2 to 3 practice problems that specifically require that formula. The goal is to associate each formula with the specific question format that triggers its use. A formula that is memorized but whose trigger is unknown is less useful than a formula whose trigger is automatic.

Stage three: integrated recall (weeks four through exam). During full practice modules, identify every formula used. After each module, note which formulas were applied correctly and which were recalled incorrectly or not at all. Targeted review of the error formulas uses the remaining preparation time efficiently.

A realistic memorization target: the formulas in this reference sheet number approximately 60 to 70 distinct items. Complete memorization of all 60 to 70 is achievable in 3 to 4 weeks with the flash card and application practice approach. Partial memorization (the 25 to 30 highest-frequency formulas) is achievable in 1 to 2 weeks and covers approximately 80 to 85 percent of all formula applications on a typical Digital SAT administration.

An efficient alternative to flash cards for some students: formula journals. Instead of cards, write each formula at the top of a page of a dedicated notebook, with the one-sentence explanation and a worked example below it. Review the journal during any brief idle moment (commuting, waiting, etc.). The act of writing the formula from memory while checking the journal builds retrieval practice more effectively than passive re-reading.

Conclusion

The Digital SAT Math section tests a specific and finite set of formulas and concepts. Every formula that appears on the test is represented in this reference sheet. Students who memorize all formulas in this reference and who understand the specific question type that triggers each formula are mathematically prepared for every formula-based question on the Digital SAT.

The reference sheet is the knowledge foundation. The strategy articles (Articles 19 through 24) provide the execution framework that converts formula knowledge into correct answers under time pressure. Together, the formula knowledge and the execution strategy constitute the complete Digital SAT Math preparation.

A well-prepared student reading this reference before the exam should feel confident recognition for every formula listed. Any formula that produces uncertainty or a memory blank during review is a formula that needs one more focused practice session. Use this recognition test as a final diagnostic before the exam day.

The complete formula reference also functions as a course completion checklist: working through the entire 25-article SAT Math series and being able to recognize every formula in this reference confirms that the preparation is comprehensive. Content gaps (topics not yet studied) correspond to formulas in this reference that are unfamiliar. Strategy gaps (execution skills not yet developed) correspond to the strategy articles (19 through 24). Formula fluency gaps (known content but slow recall) correspond to the flash card practice described in this article.

A student who can look at any formula in this reference sheet and immediately recall: (1) what the formula computes, (2) what the variables represent, (3) what question type triggers its use, and (4) at least one worked example of its application has achieved the level of formula mastery that the Digital SAT requires. This four-dimensional formula knowledge distinguishes students who merely know that a formula exists from students who can apply it fluently in context.

Extended Formula Notes: Context, Derivations, and Application Patterns

The formulas in this reference are most useful when accompanied by an understanding of their derivation and the specific question contexts that trigger each one. The following extended notes provide that context for the most important formulas.

THE SLOPE FORMULA IN DEPTH: The slope formula m = (y2 minus y1)/(x2 minus x1) quantifies the rate of change between two points. In the context of a linear model (word problem context), slope represents “the change in y per unit of x.” For example, if y represents miles driven and x represents hours, a slope of 60 means “60 miles per hour.” This contextual interpretation is tested heavily on the Digital SAT in regression interpretation questions (Article 25, Formula Domain 4).

Parallel lines have identical slopes because parallel lines have the same steepness. Perpendicular lines have slopes that are negative reciprocals because the rotation of 90 degrees transforms a slope of m into a slope of minus 1/m. A quick check: m1 times m2 = minus 1 confirms perpendicularity.

For horizontal lines, y is constant and slope = 0. For vertical lines, x is constant and slope is undefined (division by zero).

THE QUADRATIC FORMULA IN DEPTH: The quadratic formula x = (-b plus or minus sqrt(b squared minus 4ac)) / 2a solves any quadratic equation ax squared + bx + c = 0. The formula is derived by applying completing-the-square to the general quadratic, which is why the vertex x-coordinate (minus b/2a) appears in the center of the formula.

The “plus or minus” in the formula produces two solutions (the two roots). The discriminant (b squared minus 4ac) under the radical determines whether these solutions are real and distinct (discriminant greater than 0), real and equal (discriminant = 0, meaning the square root evaluates to 0 and both solutions are identical), or complex (discriminant less than 0, meaning the square root of a negative number produces an imaginary result).

Practical tip: for the Digital SAT, the quadratic formula is most useful when factoring is not obvious. If the quadratic factors easily (e.g., x squared minus 5x + 6 = (x minus 2)(x minus 3)), factoring is faster. If the quadratic does not factor over the integers (e.g., x squared + 3x + 1 = 0), the quadratic formula or Desmos zero-finding is necessary.

THE EXPONENT RULES IN DEPTH: The five exponent rules (product, quotient, power, zero, negative) follow from the definition of exponentiation as repeated multiplication.

Product rule (add exponents when multiplying same base): x cubed times x squared = (x times x times x) times (x times x) = x to the 5. Three factors times two factors = five factors.

Quotient rule (subtract exponents when dividing same base): x to the 5 divided by x squared = (x times x times x times x times x) / (x times x) = x cubed. Five factors divided by two factors = three factors.

Power rule (multiply exponents for a power of a power): (x squared) cubed = (x squared) times (x squared) times (x squared) = x to the 6. Three applications of x-squared, each contributing 2, gives 2 times 3 = 6.

Zero exponent (x to the 0 = 1): by the quotient rule, x squared / x squared = x to the (2 minus 2) = x to the 0. But any number divided by itself = 1. So x to the 0 = 1.

Negative exponent (x to the minus a = 1/x to the a): by the quotient rule, x squared / x to the 5 = x to the (2 minus 5) = x to the minus 3. But also, x squared / x to the 5 = 1/(x cubed). So x to the minus 3 = 1/(x cubed).

These derivations are not tested directly but understanding them prevents errors when applying the rules under time pressure.

THE PYTHAGOREAN THEOREM AND TRIPLES IN DEPTH: The Pythagorean theorem a squared + b squared = c squared applies only to right triangles, where c is the hypotenuse (the side opposite the right angle, always the longest side).

Pythagorean triples are integer sets (a, b, c) satisfying the theorem. Memorizing the common triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) allows instant recognition of right triangles from side lengths without computation. Any multiple of a Pythagorean triple also forms a right triangle: 6-8-10, 9-12-15, 10-24-26 are all right triangles.

On the Digital SAT, Pythagorean triples appear in coordinate geometry questions (distance between two points that happen to form a triple), in classic right triangle problems, and in the 30-60-90 and 45-45-90 triangles (which are related to specific triples).

THE DISTANCE AND MIDPOINT FORMULAS IN DEPTH: The distance formula sqrt((x2 minus x1) squared + (y2 minus y1) squared) is the Pythagorean theorem applied to the horizontal distance |x2 minus x1| and vertical distance |y2 minus y1| between two points.

The midpoint formula ((x1 + x2)/2, (y1 + y2)/2) computes the average x-coordinate and average y-coordinate of the two endpoints. A common SAT problem type: the midpoint is given along with one endpoint; find the other endpoint. Setup: ((x1 + x2)/2, (y1 + y2)/2) = (given midpoint). Solve for x2 and y2.

THE PROBABILITY FORMULAS IN DEPTH: The basic probability P(A) = favorable/total applies when all outcomes are equally likely. For dependent events (where one outcome affects another), probabilities multiply and adjust: P(A and B) = P(A) times P(B given A).

Conditional probability P(A given B) = P(A and B)/P(B) is the probability of A occurring when it is known that B has occurred. The “given B” restriction changes the sample space from all possible outcomes to only those where B occurred.

On the Digital SAT, conditional probability most commonly appears in two-way table questions. The denominator is always the total for the conditional group (a row total or column total), not the grand total. Correctly identifying the denominator is the primary skill tested in these questions.

Worked Example Applications of Key Formulas

The following worked examples show each major formula applied to a representative Digital SAT question.

EXAMPLE 1 (Slope and Perpendicular Lines): “Line l has slope 3/4. Line m is perpendicular to line l. What is the slope of line m?” Perpendicular slopes are negative reciprocals: slope of m = minus 4/3. Formula applied: slope perpendicularity rule.

EXAMPLE 2 (Quadratic Formula): “Find the roots of 2x squared + 5x minus 3 = 0.” a = 2, b = 5, c = minus 3. x = (minus 5 plus or minus sqrt(25 + 24)) / 4 = (minus 5 plus or minus sqrt(49)) / 4 = (minus 5 plus or minus 7) / 4. x = 1/2 or x = minus 3. Formula applied: quadratic formula.

EXAMPLE 3 (Vertex Formula): “Find the maximum value of f(x) = minus x squared + 4x + 1.” x-coordinate of vertex = minus(4) / (2 times minus 1) = minus 4 / minus 2 = 2. y-coordinate: f(2) = minus 4 + 8 + 1 = 5. Maximum value = 5. Formulas applied: vertex formula x = minus b/(2a), then substitution.

EXAMPLE 4 (Discriminant): “For what value of k does kx squared + 6x + 3 = 0 have exactly one real solution?” Discriminant = 0: 36 minus 4(k)(3) = 0. 36 = 12k. k = 3. Formula applied: discriminant b squared minus 4ac = 0.

EXAMPLE 5 (Percent Change): “A store reduces the price of a jacket from $80 to $60. What is the percent decrease?” (80 minus 60)/80 times 100 = 20/80 times 100 = 25 percent decrease. Formula applied: percent change = (new minus old)/old.

EXAMPLE 6 (Conditional Probability): “Of 40 students, 25 are seniors and 15 are juniors. 10 seniors and 6 juniors play sports. What is the probability that a randomly selected student plays sports given that the student is a junior?” P(sports given junior) = 6/15 = 2/5. Formula applied: P(A given B) = (count of A and B)/(count of B).

EXAMPLE 7 (Exterior Angle Theorem): “In triangle ABC, angle A = 55 degrees and angle B = 70 degrees. What is the measure of the exterior angle at C?” Exterior angle at C = angle A + angle B = 55 + 70 = 125 degrees. Formula applied: exterior angle = sum of non-adjacent interior angles.

EXAMPLE 8 (i-Power Cycle): “What is the value of i to the 53?” 53 divided by 4 = 13 remainder 1. Remainder 1 corresponds to i. So i to the 53 = i. Formula applied: i-power cycle (remainder after dividing by 4).

EXAMPLE 9 (Arc Length): “A circle has radius 9 and a central angle of 120 degrees. What is the arc length?” Arc length = (120/360) times 2 pi times 9 = (1/3) times 18 pi = 6 pi. Formula applied: arc length = (central angle/360) times circumference.

EXAMPLE 10 (Distance Formula): “What is the distance between (minus 1, 3) and (5, minus 1)?” d = sqrt((5 minus (minus 1)) squared + (minus 1 minus 3) squared) = sqrt(36 + 16) = sqrt(52) = 2 sqrt(13). Formula applied: distance formula.

These ten examples cover the most frequently tested formula applications across all four Digital SAT Math domains. Students who can work through each example in under 60 seconds have achieved a functional level of formula fluency.

For students who want to test their formula fluency more systematically: after working through all 10 examples, cover the solution and attempt each one from the formula statement alone, without the worked solution as a guide. The ability to independently apply each formula to a fresh instance of the question type (not just follow the shown solution) is the correct benchmark for exam readiness. If any example requires more than 90 seconds when attempted independently, additional practice with that formula is needed.

Formula Quick-Reference by Question Trigger

The following quick-reference table organizes formulas by the question trigger phrase or question type, enabling rapid formula selection during the exam.

“What is the slope?” or “What is the rate of change?” –> Slope formula = (y2 minus y1)/(x2 minus x1).

“What is the equation of the line?” –> Slope-intercept: y = mx + b; point-slope: y minus y1 = m(x minus x1).

“Find the roots/zeros/solutions of a quadratic” –> Quadratic formula or Desmos zero-finding or factoring.

“Find the vertex/minimum/maximum” –> Vertex x = minus b/(2a), then substitute to find y.

“How many real solutions?” –> Discriminant b squared minus 4ac: positive = 2, zero = 1, negative = 0.

“What is the remainder when divided by (x minus a)?” –> Remainder theorem: f(a).

“What is the percent change/increase/decrease?” –> Percent change = (new minus old)/old.

“Mixing two substances/solutions” –> Mixture equation: percent1 times amount1 + percent2 times amount2 = percent_total times total.

“How long to complete a task working together?” –> Combined rate = 1/a + 1/b.

“Distance between two coordinate points?” –> Distance formula = sqrt((x2 minus x1)^2 + (y2 minus y1)^2).

“Midpoint of a segment?” –> Midpoint = ((x1 + x2)/2, (y1 + y2)/2).

“Center and radius of a circle?” –> (x minus h)^2 + (y minus k)^2 = r^2.

“Length of an arc?” –> Arc length = (central angle/360) times 2 pi r.

“Area of a sector?” –> Sector area = (central angle/360) times pi r^2.

“Probability given a condition?” –> P(A given B) = (count of A and B) / (count of B).

“What does the slope represent in context?” –> Slope = average change in y per one-unit increase in x.

“What is i to the n-th power?” –> Divide n by 4, use remainder: 1 = i, 2 = minus 1, 3 = minus i, 0 = 1.

“If [dimension] is doubled/tripled, what happens to the volume?” –> Volume scales as k cubed.

“Find an angle using parallel lines” –> Corresponding angles equal; alternate interior angles equal; co-interior angles supplementary.

“What is the sum of interior angles of an n-gon?” –> (n minus 2) times 180 degrees.

This quick-reference enables the correct formula to be identified within 5 seconds of reading a question, which is the recognition-speed target for fully prepared students.

Advanced Formula Applications: When Standard Formulas Combine With Novel Contexts

Many high-difficulty Digital SAT questions take standard formulas and embed them in unusual contexts that make the formula harder to recognize. The following examples show each major formula applied in a non-standard context.

SLOPE IN A NON-STANDARD CONTEXT: Standard: “Find the slope between (2, 4) and (5, 10).” (Direct application.) Non-standard: “A linear function has f(0) = 3 and f(6) = 15. What is the rate of change?” The “rate of change” is the slope. f(0) = 3 gives point (0, 3); f(6) = 15 gives (6, 15). Slope = (15 minus 3)/(6 minus 0) = 2. Recognition challenge: the question uses “rate of change” and function notation rather than “slope” and explicit coordinate pairs.

QUADRATIC FORMULA IN A CONTEXTUAL SETTING: Standard: “Solve 2x squared minus 5x minus 3 = 0.” Non-standard: “A projectile’s height is modeled by h(t) = minus 16t squared + 32t + 48. At what time does the projectile hit the ground?” Set h(t) = 0: minus 16t squared + 32t + 48 = 0. Divide by minus 16: t squared minus 2t minus 3 = 0. Factor: (t minus 3)(t + 1) = 0. t = 3 or t = minus 1. Since t is time, t = 3 seconds. Recognition challenge: the quadratic appears as a physical model with context variables rather than a naked algebraic equation.

PERCENT CHANGE WITH MULTIPLE STEPS: Standard: “A price increases from $40 to $50. What is the percent change?” Non-standard: “After two successive 10 percent increases, a quantity is 121 percent of its original value. What is the original value if the final value is 242?” Divide 242 by 1.21: original = 200. Recognition challenge: the non-standard version requires recognizing that “two 10 percent increases” equals 1.10 times 1.10 = 1.21, and then working backward from the given final value.

DISCRIMINANT IN A TANGENCY CONTEXT: Standard: “How many real solutions does x squared minus 4x + 5 = 0 have?” Non-standard: “For what value of k does the line y = kx + 1 intersect the parabola y = x squared exactly once?” Substitute: kx + 1 = x squared, so x squared minus kx minus 1 = 0. Discriminant = k squared + 4. Set to 0: k squared = minus 4. No real solutions (the line always intersects the parabola in two real points or is tangent only if the discriminant equals 0, which requires k squared = minus 4, impossible for real k). This reveals that the line y = kx + 1 always intersects y = x squared in two distinct points for any real k. Recognition challenge: the discriminant appears as a condition for tangency after a substitution step that creates the quadratic.

CONDITIONAL PROBABILITY IN A NON-TABLE CONTEXT: Standard: “From a two-way table, find P(prefers science given junior).” Non-standard: “In a city, 40 percent of residents own dogs, and 30 percent of dog owners also own cats. What is the probability that a randomly selected resident owns both a dog and a cat?” P(dog and cat) = P(dog) times P(cat given dog) = 0.40 times 0.30 = 0.12. Recognition challenge: the conditional probability structure is present but no table is given; the question states the conditional probability directly and asks for the joint probability.

DISTANCE FORMULA IN A CIRCLE RADIUS CONTEXT: Standard: “Find the distance between (minus 1, 3) and (5, minus 1).” Non-standard: “A circle is centered at (2, minus 3) and passes through the point (8, 5). What is the radius of the circle?” The radius is the distance from the center to the given point: r = sqrt((8 minus 2) squared + (5 minus (minus 3)) squared) = sqrt(36 + 64) = sqrt(100) = 10. Recognition challenge: the distance formula appears in the context of a circle radius calculation rather than an explicit “find the distance” prompt.

EXTERIOR ANGLE THEOREM IN MULTI-STEP ANGLE CHAINS: Standard: “Triangle ABC has angles 55 degrees, 70 degrees, 55 degrees. Find the exterior angle at B.” Non-standard: “In the figure, lines m and n are parallel. Angle 1 = 60 degrees and angle 2 = 80 degrees. Find angle 3 where a triangle is formed by lines m, n, and a transversal.” The triangle formed has angles determined by the parallel line relationships. The exterior angle theorem, alternate interior angles, and the triangle angle sum work together: one step uses parallel lines to find an interior angle, the next uses the exterior angle theorem. Recognition challenge: the theorem appears as one step in a multi-step angle chain rather than a direct one-step application.

The Formula Reference in the Context of the Full 25-Article Series

Article 25 is the formula foundation article for the entire series. The earlier topic-specific articles (Articles 1 through 22) each cover the specific formulas relevant to their domain in depth. This reference article consolidates all formulas into a single searchable list for final review.

The relationship between this article and the topic articles: when a formula in this reference is applied in a specific context (like completing the square for a circle equation, which involves both the algebra completing-the-square procedure and the circle standard form), the relevant topic article provides the worked-example depth that this reference summarizes. Use this reference for memorization and recall; use the topic articles for understanding and context.

The most effective formula preparation protocol combines both: read the topic article first to understand the formula in context, then use this reference to confirm and test recall. This two-source approach builds both the conceptual understanding that makes formulas meaningful and the automatic recall that makes them fast to apply under exam pressure.

A complete Digital SAT preparation program uses this reference article as follows: during initial topic study (Articles 1 through 22), encounter formulas in context. During strategy study (Articles 19 through 24), learn how to apply formulas efficiently. During final review (this article), consolidate all formula knowledge into a single verified memorization checklist.

This approach ensures that formula memorization is not an abstract exercise but a consolidation of formulas that were first encountered in meaningful problem-solving contexts. Formulas learned in context are retained longer and recalled more reliably than formulas memorized in isolation. The twenty-five article series as a whole functions as the contextual learning engine: each topic article teaches the formula in context, and this reference article consolidates the recall. Students who complete the full series find that formula recall is largely automatic by the time they reach this reference article, because each formula has been applied multiple times in worked examples throughout the series.

The Formula Reference as a Printable Study Aid

This article functions best as a printable study aid for final review. When printing or using as a digital reference, organize the review as follows:

Pass one (unfamiliar formulas): read through the complete reference and circle or highlight any formula that requires more than 3 seconds to recall or that feels uncertain. These are the formulas that need additional flash card practice.

Pass two (application context): for each circled formula, write one sentence describing the question type that triggers its use. For example: “Sector area formula is triggered when a question asks for the area of a circular slice given the central angle and radius.”

Pass three (sample problem): for each circled formula, work through one practice problem that specifically requires that formula. The goal is to connect the formula to the problem type so that encountering the problem type triggers automatic formula recall.

Pass four (confirmation): re-read the full reference one day before the exam. For any formula that is still not automatic (takes more than 1 second to recall), write it out 5 times from memory and verify the written version against this reference. A final day-before-exam tip: do not try to learn new formulas in the 24 hours before the exam. Focus entirely on confirming what is already memorized. The exam-day benefit of trying to learn one new formula in the final 24 hours is much smaller than the cost of the extra stress and cognitive load. Trust the preparation.

This four-pass review approach, executed over the final 1 to 2 weeks before the exam, ensures that every formula in this reference is at automatic-recall fluency by exam day.

Common Formula Confusion Pairs

Several formula pairs are frequently confused with each other on the Digital SAT. The following disambiguation notes address the most common confusion pairs.

Arc length vs circumference: arc length = (fraction of circle) times circumference. Circumference = 2 pi r (the full arc length for the whole circle). When the question asks for the full boundary, use circumference. When it asks for a portion, multiply by the fraction.

Sector area vs circle area: sector area = (fraction of circle) times circle area. Circle area = pi r squared (the full area). When the question asks for the full area, use pi r squared. When it asks for a portion, multiply by the fraction.

Distance formula vs slope formula: distance uses squares and a square root (sqrt((x2-x1)^2+(y2-y1)^2)). Slope uses a simple ratio ((y2-y1)/(x2-x1)). Distance produces a length; slope produces a rate.

Mean vs sum: mean = sum/n; sum = mean times n. The confusion: students who want the mean compute the sum, and students who want the sum compute the mean. Label explicitly: “I want the mean” or “I want the sum” before computing.

P(A given B) vs P(A and B): P(A given B) has the count of A and B in the numerator and the count of B in the denominator (conditional denominator is the B group). P(A and B) has the count of A and B in the numerator and the total count in the denominator (joint probability denominator is the total).

Vertex x-coordinate vs axis of symmetry: they are the same value (x = minus b/(2a)). Some questions call it the “axis of symmetry” and ask for the equation x = [value]; others call it the vertex and ask for the coordinates (h, k). Both require x = minus b/(2a).

i squared vs i: i = sqrt(minus 1) (the imaginary unit, not a real number). i squared = minus 1 (a real number). These look similar but differ fundamentally. In FOIL expansion of complex numbers, replace every i squared with minus 1 immediately.

These confusion pairs represent the specific formula-level errors that commonly appear in Digital SAT answer mistakes. Review this disambiguation section carefully if any of these pairs cause uncertainty during practice.

For each confusion pair, the most effective disambiguation strategy is to create a single worked example that highlights the difference: apply both formulas in the pair to the same problem scenario and observe how the outputs differ. The contrast makes the distinction concrete and memorable in a way that reading the formulas in isolation does not.

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Frequently Asked Questions

Q1: What formulas does the SAT provide on the reference sheet?

The SAT provides area formulas for rectangles, triangles, and circles; circumference of a circle; volume formulas for rectangular prisms, cylinders, spheres, cones, and pyramids; the Pythagorean theorem; special right triangle ratios (30-60-90 and 45-45-90); and the degree measures for circles (360 degrees) and triangles (180 degrees). Everything else in this article must be memorized. A practical implication: even for the provided formulas (like the volume of a cone = (1/3) pi r squared h), it is worth having them memorized because checking the reference sheet takes 5 to 10 seconds. Over the course of a module, memorizing even the provided formulas saves a cumulative 30 to 60 seconds. The reference sheet is a safety net, not a replacement for memorization. The most frequently used provided formulas: area of a circle (pi r squared) and the Pythagorean theorem (a squared + b squared = c squared) appear in many questions. These are worth memorizing first among the provided formulas, followed by the special right triangle ratios (which appear surprisingly often on coordinate geometry questions).

Q2: How many formulas do I need to memorize for the Digital SAT Math section?

Approximately 60 to 70 distinct formulas and rules. The 25 to 30 highest-frequency formulas (slope, quadratic formula, vertex formula, discriminant, exponent rules, percent change, conditional probability, distance formula, midpoint formula, and the key rules) cover approximately 80 to 85 percent of all formula applications on a typical administration. Complete memorization of all formulas in this reference is the goal for students targeting 700 and above. Students targeting 600 to 650 can achieve significant score improvements by memorizing the first tier (highest frequency) formulas, which includes slope, the quadratic formula, vertex formula, discriminant, percent change, and the five exponent rules. These first-tier formulas appear on nearly every module. A useful benchmark: a student who needs more than 3 seconds to recall any first-tier formula has not yet memorized it to automaticity. The goal is instant recall (under 1 second) for all first-tier formulas, which requires repeated retrieval practice (flash cards) beyond initial learning.

Q3: Which domain has the most formulas to memorize?

Algebra has the most formulas (slope, intercepts, quadratic formula, vertex formula, discriminant, exponent rules, absolute value), followed by Advanced Math (polynomial theorems, complex numbers, function notation, exponential models). Geometry formulas are partly provided by the SAT reference sheet, reducing the memorization burden. Statistics formulas (mean, probability, conditional probability) are fewer in number but conceptually important. A useful prioritization: since Algebra and Advanced Math together account for approximately 70 percent of Digital SAT Math questions, the formulas in those domains deserve the most memorization effort. Geometry formulas are supplemented by the reference sheet. Statistics formulas are fewer in number. The time allocation for memorization should roughly mirror the domain weight: more time on Algebra and Advanced Math formulas, less time on Geometry (partly provided) and Statistics (fewer formulas). An important note about the Geometry and Trigonometry domain: the trigonometric ratios (SOH-CAH-TOA), the complementary angle identity (sin x = cos(90-x)), and the radian-degree conversion are NOT on the SAT reference sheet and must be memorized. Students who neglect trigonometry formulas and rely on the reference sheet will be unprepared for trig questions.

Q4: What is the most commonly missed formula on the Digital SAT?

The vertex formula x = minus b/(2a) is one of the most commonly missed because of the negative sign. Students who know the formula sometimes forget the negative, computing b/(2a) instead of minus b/(2a). This produces the wrong vertex x-coordinate and the wrong answer. Writing the negative sign first before substituting values prevents this specific error. Close competitors for most-commonly-missed formula: the conditional probability denominator (using the grand total instead of the row or column total) and the percent change denominator (using the new value instead of the old value). Both involve correctly identifying which quantity goes in the denominator, and both benefit from the labeling habits in Article 23. A less obvious commonly missed formula: the slope formula with a sign error. Students compute (y1 minus y2)/(x2 minus x1) (mixing which numerator is subtracted from which) and get the negative of the correct slope. The fix: always write the formula explicitly as (y2 minus y1)/(x2 minus x1) and subtract consistently.

Q5: How do I remember the difference between combinations and permutations?

Combinations (C) = choices where order does not matter (choosing a committee, selecting items from a group). Permutations (P) = choices where order matters (arranging items in sequence, creating passwords, ranking). The mnemonic: “Combinations are for Collections (order irrelevant); Permutations are for Positions (order matters).” Combinations always produce a smaller or equal count than permutations for the same n and k. On the Digital SAT, most counting problems that appear are either direct counting-principle applications (m times n) or combinations (how many ways to choose a group). Pure permutation questions (ordered arrangements) appear less frequently. When uncertain whether a problem requires combinations or permutations, ask: “does switching the order of the selected items give a different result?” If yes, use permutations. If no, use combinations. A concrete test: “choose 3 students from 10 for a committee” - does switching which student is “first” vs “second” matter? No (a committee is a committee regardless of order). Use combinations: C(10,3) = 120. “Arrange 3 books from 10 on a shelf” - does the order of the books on the shelf matter? Yes. Use permutations: P(10,3) = 720.

Q6: When does the i-power cycle apply?

Any time a question asks for i to a power greater than 4. Divide the exponent by 4 and use the remainder: remainder 1 = i, remainder 2 = minus 1, remainder 3 = minus i, remainder 0 (or 4) = 1. Example: i to the 47 = i to the (44 + 3) = (i to the 44) times (i cubed) = 1 times (minus i) = minus i. The cycle applies because i to the 4 = 1, so every 4 powers return to 1. A clean verification: i to the 4 = (i squared) squared = (minus 1) squared = 1. Confirmed. For i to the 8: same calculation gives 1 again. The cycle repeating every 4 powers is provable directly from i squared = minus 1.

Q7: What is the practical difference between mean, median, and mode in SAT questions?

The SAT tests mean most heavily (computing it, working backward from it, finding a missing value when the mean is given). Median questions typically ask about the effect of adding or removing a value; the median of an odd-count data set is the middle value after sorting. Mode questions are relatively rare and typically ask which value appears most frequently. A specific SAT trap: adding a value that is larger than all existing values shifts the mean upward but may not change the median if the new value is not in the middle. A key formula for working backward from mean: if mean = M and the number of values = n, then sum = M times n. If a new value v is added, the new mean = (sum + v)/(n + 1). This backward-calculation technique appears in Digital SAT questions that give the mean and ask for a missing value. Standard deviation, while tested conceptually (which distribution is more spread?), is rarely tested computationally on the Digital SAT. Students should know that standard deviation measures spread and that it increases when data points move farther from the mean.

Q8: How do I remember the area formulas that are NOT provided by the SAT?

The main area formula not provided is the surface area of 3D solids (rectangular prism SA = 2(lw + lh + wh); cylinder SA = 2 pi r squared + 2 pi r h; sphere SA = 4 pi r squared). The key insight: surface area is the total area of all faces. For a rectangular prism, there are 3 pairs of opposite faces: lw (top and bottom), lh (front and back), wh (left and right). Summing all 6 faces gives the formula. For a cylinder: the surface area has two circular ends (pi r squared each = 2 pi r squared total) plus the lateral surface (which unrolls into a rectangle with height h and width equal to the circumference 2 pi r, giving 2 pi r h area). Total: 2 pi r squared + 2 pi r h. For a sphere: SA = 4 pi r squared (the SAT provides the sphere volume formula but not this surface area formula; it appears less frequently but is worth memorizing for completeness). An important distinction: the SAT provides volume formulas but not surface area formulas for the same solid shapes. When a question specifically asks about surface area rather than volume, the reference sheet will not help. Knowing which questions are about surface area (they typically use the phrase “surface area” or ask about the total outer surface) prevents the confusion of applying volume formulas to surface area questions.

Q9: What is the difference between arc length and sector area?

Arc length is the distance along the curved boundary of a circular sector (a one-dimensional measurement in the same units as the radius). Sector area is the area of the “pie slice” portion of the circle (a two-dimensional measurement in square units). Both use the same fraction of the full circle (central angle / 360), applied to circumference for arc length and to total circle area for sector area. A memory aid: arc length uses 2 pi r (the full circumference); sector area uses pi r squared (the full circle area). Both use the same fractional multiplier. The question type determines which to apply: “how long is the curved edge?” uses arc length; “how much area does the slice cover?” uses sector area. An additional note: radians simplify both formulas. Arc length in radians: L = r times theta (where theta is in radians). Sector area in radians: A = (1/2) r squared times theta. Both radians formulas eliminate the 360-degree division and are cleaner for calculation. Students comfortable with radians can use either form.

Q10: How do I apply the complementary angle trig identity?

sin(theta) = cos(90 minus theta) and cos(theta) = sin(90 minus theta). If a question states “sin(40 degrees) = 0.643” and asks “what is cos(50 degrees)?”, the answer is 0.643 because 50 = 90 minus 40. The identity follows from the structure of a right triangle: the two acute angles are complementary (sum to 90), and sin of one equals cos of the other. This identity is one of the most directly testable trig facts on the Digital SAT because it requires knowing the relationship, not performing any calculation. The question often provides one trig value and asks for a trig value of the complementary angle in a deliberately confusing format. Recognizing the “90 minus” relationship immediately resolves the question without any computation. The identity extends: tan(theta) = cot(90 minus theta) (tangent of one angle equals the cotangent of the complement), though cotangent appears rarely on the Digital SAT. The primary identity to memorize is sin(theta) = cos(90 minus theta).

Q11: When should I use the distance formula versus the Pythagorean theorem?

They are equivalent. The distance formula for (x1, y1) to (x2, y2) is derived by creating a right triangle with legs

x2 minus x1

and

y2 minus y1

and applying the Pythagorean theorem. Use whichever framing is more natural for the question context. For explicit coordinate pairs in the plane, the distance formula is more direct. For right triangle configurations drawn in the problem, the Pythagorean theorem may be more natural. Desmos integration: for questions where only the distance value is needed (not the formula), plot the two points on Desmos and click each point to confirm coordinates, or use the Desmos distance calculator by typing sqrt((x2-x1)^2+(y2-y1)^2) with specific values substituted. A practical note: recognizing Pythagorean triples (3-4-5, 5-12-13, 8-15-17) in coordinate geometry problems allows instant distance identification without computation. If the legs of the coordinate right triangle are 3 and 4, the hypotenuse (distance) is 5.

Q12: What is the most efficient way to find the y-intercept of a line given two points?

Method 1: find the slope using (y2 minus y1)/(x2 minus x1), then substitute one point and the slope into y = mx + b and solve for b. Method 2: substitute both points as simultaneous equations (y1 = m times x1 + b and y2 = m times x2 + b) and solve the system. Method 3 (Desmos): enter the two points in a Desmos table and the regression line y1 ~ mx1 + b gives the slope and intercept directly. Method 1 is the fastest paper-based approach for most students. The calculation: find m, then b = y1 minus m times x1. For example, given points (2, 5) and (4, 9): m = (9 minus 5)/(4 minus 2) = 2. b = 5 minus 2 times 2 = 1. Equation: y = 2x + 1. Total time: approximately 30 seconds.

Q13: How do I identify whether a data set has a linear or exponential relationship?

The two-test: compute first differences (y2 minus y1, y3 minus y2, etc.) and ratios (y2/y1, y3/y2, etc.). If first differences are constant, the relationship is linear. If ratios are constant, the relationship is exponential. For data that is approximately but not exactly linear or exponential, use Desmos regression to compare R-squared values for both models. A common Digital SAT question type: a table of values is provided, and the question asks which type of model (linear, quadratic, or exponential) best fits. After the two-test rules out linear and exponential, a constant second difference (difference of differences) indicates quadratic. Desmos regression confirms which model has the better R-squared fit. A practical shortcut for quick model identification: if the first y-value is much smaller than the last y-value and the data seems to be accelerating (growing faster and faster), the relationship is likely exponential. If the data grows at roughly the same rate throughout, it is likely linear. This visual identification from the table values takes 3 to 5 seconds and eliminates one or two models before applying the two-test.

Q14: What is conditional probability and how does it differ from simple probability?

Simple probability: P(A) = favorable outcomes / total outcomes. Conditional probability: P(A given B) = (count of A AND B) / (count of B). The “given B” restricts the sample space to only those outcomes where B occurs; B becomes the new denominator. Example from a two-way table: if the table has 80 total students, 30 juniors, and 12 juniors who prefer science, then P(science) = 12/80 = 15 percent, but P(science given junior) = 12/30 = 40 percent. The denominator changes from all students to just juniors. The “given” language is the trigger: any question containing “given that,” “if it is known that,” or “among those who” is asking for conditional probability. The denominator is the total for the “given” group, not the grand total. The most common conditional probability error on the Digital SAT: using the wrong denominator. Prevention: before writing the probability fraction, explicitly ask “what is the conditional group?” and write that group’s total as the denominator. Then find the numerator (the count of the target attribute within that group). This two-step protocol prevents the grand-total denominator error.

Q15: How does the counting principle relate to combinations and permutations?

The counting principle (m times n for two sequential events) is the foundational principle. Permutations apply the counting principle to ordered selections: choosing 3 from 10 in order = 10 times 9 times 8 = P(10, 3). Combinations divide permutations by k! (the number of orderings of the k selected items that are considered the same): C(10, 3) = P(10, 3) / 3! = 720/6 = 120. Use the counting principle directly for simple sequential choice problems; use combinations when order does not matter. On the Digital SAT, the counting principle appears in questions like “a restaurant offers 4 appetizers, 5 main courses, and 3 desserts; how many different three-course meals are possible?” (answer: 4 times 5 times 3 = 60). Combinations appear in questions like “a committee of 3 is chosen from 8 candidates; how many committees are possible?” (answer: C(8,3) = 56). For combination calculation without the formula: C(n, k) = n! / (k! times (n-k)!). For C(8, 3): 8 times 7 times 6 / (3 times 2 times 1) = 336/6 = 56. The numerator is the top k terms of n! (8, 7, 6 for k=3); the denominator is k!.

Q16: Why does the SAT test the vertex formula and discriminant specifically?

The vertex and discriminant are the two key properties of a quadratic function that determine its graphical and solution behavior. The vertex tells you the maximum or minimum value and the axis of symmetry. The discriminant tells you how many real solutions exist. Together, they fully characterize a quadratic without requiring full factoring or the quadratic formula. The SAT tests them specifically because they require understanding the relationship between the algebraic formula and the geometric/solution properties. For the vertex formula: the connection between a quadratic’s algebraic form (ax squared + bx + c) and its geometric vertex (minimum or maximum point) is tested in both directions: “given the formula, find the vertex” and “given the vertex, find the formula.” Both directions require understanding x = minus b/(2a). For the discriminant: the SAT specifically tests whether students understand the qualitative behavior of a quadratic (how many real solutions) without requiring them to solve it. The discriminant converts a question about solutions into a simple arithmetic check, which tests conceptual understanding rather than procedural solving ability.

Q17: How do I handle problems that combine multiple formulas?

Most hard Digital SAT questions require chaining two or three formulas in sequence. For example: finding the area of a sector requires (1) finding the central angle, (2) applying the sector area formula using that angle. Finding the distance between two intersection points requires (1) solving the system to find the intersection points, (2) applying the distance formula to those points. The key skill is sequencing: identify what each step produces and which formula takes that product as an input. Practice identifying the “output-input chain” in multi-step problems: the output of step 1 is the input to step 2. This systematic chain-identification habit prevents the common error of applying formulas in the wrong order or skipping a step. A useful pre-solve strategy for multi-formula questions: before computing, map out the chain on scratch paper: “Step 1: find [X] using [formula]. Step 2: use [X] in [formula] to find [Y]. Final answer: [Y].” This 15-second map prevents getting lost mid-problem.

Q18: What is the reading-time cost of opening the SAT reference sheet?

Opening the reference sheet in Bluebook takes approximately 5 to 10 seconds (touch/click to open, scan for the needed formula, close). If a formula is memorized, recall takes approximately 1 to 2 seconds. For the high-frequency formulas (slope, quadratic formula, discriminant, vertex), memorization is significantly faster than reference sheet lookup and is strongly recommended. For the rarely tested volume formulas (cone, sphere, pyramid), using the reference sheet when needed is efficient enough that memorization may not be worth the preparation time. An interesting point about the reference sheet: its presence actually supports the SAT’s emphasis on higher-order reasoning. By providing basic geometric formulas, the SAT signals that memorizing those formulas is not the tested skill; applying them in complex contexts is.

Q19: How should I prioritize formula memorization if I have limited preparation time?

Memorize in this order, based on Digital SAT frequency: First tier (highest frequency, memorize first): slope formula, slope-intercept form, quadratic formula, vertex formula, discriminant, percent change, conditional probability formula, distance formula, angle relationships (supplementary, vertical, exterior angle theorem). Second tier: midpoint formula, circle equation, exponent rules, mean formula, arc length, sector area, absolute value inequality cases. Third tier: combinations/permutations, i-power cycle, surface area formulas, sum of cubes, sampling/margin of error concepts. Students with 1 to 2 weeks should focus on first tier. Students with 3 to 4 weeks should achieve second tier as well. Full preparation covers all three tiers. A practical allocation: the first tier contains approximately 15 to 20 formulas and requires 1 to 2 weeks of flash card study plus application practice to memorize. The second tier adds 10 to 15 more formulas. The third tier adds the remaining 10 to 15. Students with access to this complete reference sheet and 4 or more weeks before the exam can achieve full formula coverage through systematic daily practice.

Q20: Is there any formula that appears almost every Digital SAT administration?

Yes. The slope formula, the quadratic formula or discriminant, percent change, and conditional probability from a two-way table appear on nearly every Digital SAT Math administration. These four formula areas are the highest-priority memorization targets for any student. A student who can instantly recall and correctly apply these four formulas has addressed the most frequently appearing formula-based questions across all Digital SAT Math administrations. A student who additionally memorizes the vertex formula, the distance formula, the arc length formula, the exterior angle theorem, and the exponent rules has covered essentially all high-frequency formula applications. These nine formula areas together account for approximately 70 to 80 percent of all formula applications on a typical Digital SAT Math section.