SAT Math Formula Cheat Sheet — Every Formula by Topic (Free)

What to Memorize vs. What's Given

The digital SAT gives you a built-in reference sheet on every Math question — but it is geometry only. Everything in Algebra, Advanced Math, and Data Analysis you must know from memory. This is the single biggest source of avoidable mistakes, so start here.

Given to you (don't memorize)

Area & circumference of a circle
Area of rectangles and triangles
Volumes (box, cylinder, sphere, cone, pyramid)
Pythagorean theorem
Special right triangles (30-60-90, 45-45-90)
360° / 2π in a circle, 180° in a triangle

Must memorize (not given)

Slope & all three line forms
Quadratic formula & discriminant
Percent change, distance, midpoint
Exponential growth, compound interest
Mean / median, probability, averages
SOH-CAH-TOA & trig identities

About 70% of Math questions are Algebra and Advanced Math — none of those formulas are on the reference sheet. The formulas below marked with everyday topics are the ones worth drilling until they're automatic.

Algebra

Slope

m = \dfrac{y_2 - y_1}{x_2 - x_1}

Rise over run. Positive slope goes up left-to-right; negative goes down.

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept (where the line crosses the y-axis).

Point-Slope Form

y - y_1 = m(x - x_1)

Use when you know a point and the slope but not the y-intercept.

Standard Form

Ax + By = C

A, B, and C are integers. Slope = −A/B; y-intercept = C/B.

Average Rate of Change

\dfrac{f(b) - f(a)}{b - a}

The slope between two points on a curve — a very common 'interpret the model' question.

Direct & Inverse Variation

\text{Direct: } y = kx

\text{Inverse: } y = \dfrac{k}{x}

k is the constant of proportionality. 'Varies directly' = multiply; 'inversely' = divide.

Percent Change

\%\text{ Change} = \dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100

Positive = increase; negative = decrease. Always divide by the original.

Percent of

\text{Part} = \text{Percent} \times \text{Whole}

Convert percent to decimal first: 35% = 0.35.

Distance = Rate × Time

d = r \times t

Rearrange as needed: r = d/t, t = d/r.

Simple Interest

I = P \times r \times t

P = principal, r = annual rate (decimal), t = time in years.

Linear equations are the most common algebra question type. Master all three line forms (slope-intercept, point-slope, standard) and be ready to interpret slope and intercept in a real-world context.

Quadratics

Standard Form

y = ax^2 + bx + c

Vertex x-coordinate = −b / (2a). Opens up if a > 0, down if a < 0.

Vertex Form

y = a(x - h)^2 + k

Vertex is at (h, k). a determines width and direction.

Factored Form

y = a(x - r_1)(x - r_2)

r₁ and r₂ are the x-intercepts (roots/zeros).

Quadratic Formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Use when factoring is difficult. Set the equation to 0 first.

Discriminant

b^2 - 4ac

> 0: two real roots · = 0: one real root · < 0: no real roots.

Sum & Product of Roots

\text{Sum} = -\dfrac{b}{a}

\text{Product} = \dfrac{c}{a}

A shortcut when a question asks about roots without solving the full equation.

Completing the Square

ax^2 + bx + c \rightarrow a\left(x + \tfrac{b}{2a}\right)^2 + \left(c - \tfrac{b^2}{4a}\right)

Convert to vertex form. Add (b/2)² to both sides when a = 1.

Difference of Squares

a^2 - b^2 = (a + b)(a - b)

One of the most-tested factoring patterns — recognize it on sight.

The discriminant (b² − 4ac) is extremely common. If a question says "no real solutions," you know b² − 4ac < 0. If it says "exactly one solution," you know b² − 4ac = 0.

Exponential Growth, Decay & Interest

Exponential Growth

y = a(1 + r)^t

a = starting amount, r = growth rate (decimal), t = time. Each period multiplies by (1 + r).

Exponential Decay

y = a(1 - r)^t

Same form, but the quantity shrinks. Used for depreciation and half-life style problems.

General Exponential Model

y = a \cdot b^x

b > 1 means growth; 0 < b < 1 means decay. b is the per-step multiplier.

Compound Interest

A = P\left(1 + \dfrac{r}{n}\right)^{nt}

n = times compounded per year. For annual compounding, n = 1 → A = P(1 + r)ᵗ.

None of these are on the reference sheet, yet exponential models appear on almost every test. The most common task is matching a scenario ("a population grows 4% per year") to the correct equation.

Geometry

Area of Triangle

A = \dfrac{1}{2} \times \text{base} \times \text{height}

Area of Rectangle

A = \text{length} \times \text{width}

Area of Circle

A = \pi r^2

Circumference of Circle

C = 2\pi r = \pi d

Area of Trapezoid

A = \dfrac{1}{2}(b_1 + b_2) \times h

b₁ and b₂ are the parallel bases.

Volume of Rectangular Prism

V = l \times w \times h

Volume of Cylinder

V = \pi r^2 h

Volume of Cone

V = \dfrac{1}{3}\pi r^2 h

Volume of Sphere

V = \dfrac{4}{3}\pi r^3

Pythagorean Theorem

a^2 + b^2 = c^2

c is the hypotenuse (longest side, opposite the right angle).

Special Right Triangle 30-60-90

\text{Sides: } x,\ x\sqrt{3},\ 2x

Short leg : long leg : hypotenuse.

Special Right Triangle 45-45-90

\text{Sides: } x,\ x,\ x\sqrt{2}

Both legs equal; hypotenuse = leg × √2.

Sum of Interior Angles

(n - 2) \times 180^\circ

n = number of sides. Each angle of a regular polygon = (n − 2)·180° / n.

Arc Length

\dfrac{\theta}{360} \times 2\pi r

θ is the central angle in degrees.

Sector Area

\dfrac{\theta}{360} \times \pi r^2

The 'pizza slice' area for central angle θ.

Common Pythagorean triples on the test: 3-4-5, 5-12-13, 8-15-17, and multiples (6-8-10, 9-12-15). Recognizing them instantly saves you from the full theorem.

Coordinate Geometry

Midpoint Formula

M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)

Distance Formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Derived from the Pythagorean theorem.

Equation of a Circle

(x - h)^2 + (y - k)^2 = r^2

Center (h, k), radius r. Complete the square to get here.

Parallel Lines

\text{Same slope } (m_1 = m_2)

\text{Different } y\text{-intercepts}

Parallel lines never intersect.

Perpendicular Lines

m_1 \times m_2 = -1

Slopes are negative reciprocals — e.g., 2 and −½.

Statistics & Data Analysis

Mean (Average)

\text{Mean} = \dfrac{\text{Sum of values}}{\text{Number of values}}

Also: Sum = Mean × Count. Use this when a value is missing.

Median

Middle value of ordered data set

For an even count, average the two middle values. Resistant to outliers.

Mode

Most frequently occurring value

A set can have multiple modes or none.

Range

\text{Range} = \text{Max} - \text{Min}

Weighted Average

\dfrac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}}

Use when groups have different sizes — don't just average the averages.

Standard Deviation

Spread of data around the mean

You won't compute it, but know: larger SD = more spread out data.

Line of Best Fit

\hat{y} = mx + b

Slope = predicted rate of change. Be careful predicting far outside the data range.

Margin of Error

\text{Estimate} \pm \text{margin}

A larger random sample narrows the interval; only random samples justify generalizing.

When a table or graph is provided, read the title, axis labels, and units before the data. Most statistics questions reward careful reading, not heavy computation.

Probability

Basic Probability

P(\text{event}) = \dfrac{\text{Favorable}}{\text{Total}}

Complement Rule

P(\text{not } A) = 1 - P(A)

P(A and B) — Independent

P(A \cap B) = P(A) \times P(B)

Only when events are independent (one doesn't affect the other).

P(A or B)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Conditional Probability

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}

Read as 'probability of A given B.' Divide by the relevant row/column total.

Expected Value

E = \sum [x \times P(x)]

Sum of each outcome multiplied by its probability.

Two-way table questions are the most common probability type. Decide first whether the denominator is the grand total (joint probability) or a single row/column total (conditional probability).

Exponents & Radicals

Product Rule

a^m \times a^n = a^{m+n}

Quotient Rule

\dfrac{a^m}{a^n} = a^{m-n}

Power Rule

(a^m)^n = a^{mn}

Zero Exponent

a^0 = 1 \quad (a \neq 0)

Negative Exponent

a^{-n} = \dfrac{1}{a^n}

Fractional Exponent

a^{m/n} = \sqrt[n]{a^m}

E.g., 8^(2/3) = (∛8)² = 4.

Square Root Rules

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

Scientific Notation

a \times 10^n \quad (1 \le |a| < 10)

Trigonometry

SOH-CAH-TOA

\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}

\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}

\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

Reciprocal Identities

\csc\theta = \dfrac{1}{\sin\theta}

\sec\theta = \dfrac{1}{\cos\theta}

\cot\theta = \dfrac{1}{\tan\theta}

Complementary Angles

\sin\theta = \cos(90^\circ - \theta)

\cos\theta = \sin(90^\circ - \theta)

Co-function identity: sin and cos of complementary angles are equal.

Unit Circle Key Values

\sin 30^\circ = \tfrac{1}{2},\ \cos 30^\circ = \tfrac{\sqrt{3}}{2}

\sin 45^\circ = \tfrac{\sqrt{2}}{2},\ \cos 45^\circ = \tfrac{\sqrt{2}}{2}

\sin 60^\circ = \tfrac{\sqrt{3}}{2},\ \cos 60^\circ = \tfrac{1}{2}

Radian-Degree Conversion

\text{Radians} = \text{Degrees} \times \dfrac{\pi}{180}

\text{Degrees} = \text{Radians} \times \dfrac{180}{\pi}

Trigonometry is roughly 5–10% of Math questions. The most common type asks you to find a missing side with SOH-CAH-TOA or to apply sin θ = cos(90° − θ). Know these two cold.

Formulas Given on Test Day

The test provides this reference sheet on every Math question. You do not need to memorize these — but knowing them saves you from opening the panel mid-question:

✓Area of circle:

A = \pi r^2

✓Circumference:

C = 2\pi r

✓Area of rectangle:

A = lw

✓Area of triangle:

A = \tfrac{1}{2}bh

✓Pythagorean theorem:

a^2 + b^2 = c^2

✓Special right triangles:

\text{30-60-90},\ \text{45-45-90}

✓Volume of rectangular box:

V = lwh

✓Volume of cylinder:

V = \pi r^2 h

✓Volume of sphere:

V = \tfrac{4}{3}\pi r^3

✓Volume of cone:

V = \tfrac{1}{3}\pi r^2 h

✓Volume of pyramid:

V = \tfrac{1}{3}lwh

✓Arc/angle facts:

360^\circ,\ 2\pi\text{ in a circle},\ 180^\circ\text{ in a triangle}

The formulas not on the reference sheet (and most commonly tested) are: the quadratic formula, slope and line forms, distance, midpoint, percent change, exponential growth/interest, and every statistics formula. Those you must memorize.