AI-Math - SAT Math Reference Sheet Formulas Cheat Sheet

What the SAT gives you — and what it does not

On the digital SAT, a reference sheet is available on every math question, one tap away in the testing app. It is short: five area and perimeter facts, the Pythagorean theorem, two special right triangles, five volume formulas, and three angle facts. That is the complete list.

Nothing algebraic is given. The quadratic formula, slope, the equation of a circle, exponential growth, percent change, averages, probability, arc length, and every trigonometric relationship are all assumed knowledge. Roughly 70% of the math section is Algebra and Advanced Math, so the reference sheet covers almost none of what you will actually be tested on.

A built-in graphing calculator is also available on every math question, and you may bring your own approved calculator. That changes the calculus of memorization: you rarely need to recall a numeric value, but you do need to recall which relationship applies, because the calculator cannot choose the formula for you.

Given on the SAT reference sheet — plane geometry

Area of a circle

$A = \pi r^2$

Given on the test. $r$ is the radius, not the diameter — halve the diameter first, which is the most common slip on circle questions.

Circumference of a circle

$C = 2\pi r = \pi d$

Given on the test. Distance around a circle. Combine with $A = \pi r^2$ when a question gives you one and asks for the other.

Area of a rectangle

$A = \ell w$

Given on the test. Length times width; a square is the case $\ell = w$ .

Area of a triangle

$A = \tfrac{1}{2} b h$

Given on the test. $h$ is the perpendicular height to the chosen base, not the slanted side.

Pythagorean theorem

$c^2 = a^2 + b^2$

Given on the test. Right triangles only, with $c$ the hypotenuse. Memorize the common triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25 to skip the arithmetic.

30-60-90 triangle

$x \;:\; x\sqrt{3} \;:\; 2x$

Given on the test. The side opposite 30 degrees is $x$ , opposite 60 degrees is $x\sqrt{3}$ , and the hypotenuse is $2x$ . Half of an equilateral triangle.

45-45-90 triangle

$s \;:\; s \;:\; s\sqrt{2}$

Given on the test. Both legs equal $s$ and the hypotenuse is $s\sqrt{2}$ . Appears whenever a square is cut along a diagonal.

Sum of angles in a triangle

$\alpha + \beta + \gamma = 180^\circ$

Given on the test. Extends to any polygon through the interior angle sum formula, which is not given.

Given on the SAT reference sheet — volume and angle measure

Volume of a rectangular box

$V = \ell w h$

Given on the test. Length times width times height.

Volume of a cylinder

$V = \pi r^2 h$

Given on the test. Base area times height. Every prism follows the same base-times-height pattern.

Volume of a sphere

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

Given on the test. A hemisphere is half of this, which the reference sheet does not state.

Volume of a cone

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

Given on the test. Exactly one third of the cylinder with the same base and height.

Volume of a pyramid

$V = \tfrac{1}{3}\ell w h$

Given on the test. One third of the box with the same base and height; $h$ is the vertical height, not the slant height.

Degrees and radians in a circle

$360^\circ = 2\pi \text{ radians}$

Given on the test as two separate facts. The conversion itself — multiply degrees by $\pi/180$ — is not given.

Not given — lines and linear equations

Slope between two points

$m = \frac{y_2 - y_1}{x_2 - x_1}$

The single most used formula on the SAT math section. Rise over run, and it is the constant rate of change in every linear word problem.

Slope-intercept form

$y = mx + b$

Slope $m$ and $y$ -intercept $b$ . In context, $m$ is the per-unit rate and $b$ is the starting or fixed amount.

Point-slope form

$y - y_1 = m(x - x_1)$

Fastest way to write a line through a known point with a known slope, with no algebra needed to find $b$ .

Standard form and its slope

$ax + by = c \;\Rightarrow\; m = -\frac{a}{b}$

Read the slope straight off standard form without rearranging. Saves time on parallel and perpendicular questions.

Parallel and perpendicular slopes

$m_1 = m_2 \quad\text{vs}\quad m_1 m_2 = -1$

Parallel lines share a slope; perpendicular slopes are negative reciprocals. A horizontal line and a vertical line are the special case.

Distance between two points

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

The Pythagorean theorem applied to the coordinate plane. Not on the reference sheet, so memorize it.

Midpoint of a segment

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

Average the coordinates. Reverse it to find an endpoint when the midpoint and one end are given.

Number of solutions to a linear system

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

For $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ : this condition means no solution (parallel lines). If all three ratios are equal there are infinitely many; otherwise exactly one.

Not given — quadratics and polynomials

Quadratic formula

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

Solves $ax^2 + bx + c = 0$ every time, including when factoring fails. Not on the reference sheet — this is the most important formula to memorize for the SAT.

Discriminant

$\Delta = b^2 - 4ac$

Positive means two real solutions, zero means exactly one, negative means none. Questions asking how many times a parabola crosses the $x$ -axis are discriminant questions.

Vertex form of a parabola

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

The vertex is $(h, k)$ , read directly. This is the form to convert to whenever a question asks for a maximum or minimum value.

Axis of symmetry and vertex from standard form

$x = -\frac{b}{2a}$

The $x$ -coordinate of the vertex of $y = ax^2 + bx + c$ . Substitute back to get the $y$ -coordinate.

Sum and product of roots

$r_1 + r_2 = -\frac{b}{a}, \quad r_1 r_2 = \frac{c}{a}$

Answers questions about the roots without solving for them. The sum also gives the axis of symmetry, since the vertex sits midway between the roots.

Factored form and zeros

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

The zeros are $r_1$ and $r_2$ . Any question about where a graph crosses the $x$ -axis wants this form.

Difference of squares

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

The most frequently useful factoring identity on the test, and the fast route through many rational expressions.

Perfect square trinomials

$(a \pm b)^2 = a^2 \pm 2ab + b^2$

Recognizing these instantly speeds up completing the square and converting to vertex form.

Not given — exponents, radicals and growth

Exponent rules

$x^a x^b = x^{a+b}, \quad \frac{x^a}{x^b} = x^{a-b}, \quad (x^a)^b = x^{ab}$

Multiply means add exponents, divide means subtract, power of a power means multiply. Nearly every exponent question is one of these three.

Negative and zero exponents

$x^{-a} = \frac{1}{x^a}, \quad x^0 = 1 \;(x \neq 0)$

A negative exponent means reciprocal, not a negative value. Anything nonzero to the power zero is 1.

Fractional exponents and radicals

$x^{a/b} = \sqrt[b]{x^a}$

Converts between radical and exponent notation. The SAT often states a rule in one form and asks for the answer in the other.

Exponential growth and decay

$y = a(1 + r)^t \quad\text{or}\quad y = a(1 - r)^t$

$a$ is the initial amount, $r$ the rate per period as a decimal, $t$ the number of periods. Growth uses plus, decay uses minus.

Growth with a general multiplier

$y = a \cdot b^{t/k}$

Use when a quantity multiplies by $b$ every $k$ time units — doubling every 6 hours is $b = 2$ , $k = 6$ . This form handles most SAT half-life and doubling questions.

Compound interest

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

$P$ is the principal, $r$ the annual rate, $n$ the number of compoundings per year, $t$ years. Watch for quarterly ( $n = 4$ ) and monthly ( $n = 12$ ).

Not given — percentages, ratios and statistics

Percent of a whole

$\text{part} = \frac{\text{percent}}{100} \times \text{whole}$

Rearrange for whichever quantity is missing. Translating "of" as multiplication and "is" as equals converts most percent word problems directly.

Percent change

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

The denominator is always the original value. Reversing it is the classic SAT percent trap.

Successive percent changes

$\text{final} = \text{original} \times (1 \pm r_1)(1 \pm r_2)$

Percent changes multiply and never add: 20% off then 10% off leaves $0.8 \times 0.9 = 0.72$ , a 28% total discount.

Mean (average)

$\bar{x} = \frac{\text{sum of values}}{\text{number of values}}$

Most SAT average questions are really about the sum: multiply the mean by the count to recover it, then adjust.

Direct and inverse variation

$y = kx \quad\text{vs}\quad y = \frac{k}{x}$

Direct variation means the ratio $y/x$ is constant; inverse variation means the product $xy$ is constant. Find $k$ from the given pair, then apply.

Probability of an event

$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

On the SAT this is usually read from a two-way table. Check carefully whether the question restricts you to one row or column.

Interpreting standard deviation

$\text{smaller } \sigma \Rightarrow \text{values closer to the mean}$

The SAT never asks you to compute a standard deviation, only to compare spread between two data sets. No formula needed, only the concept.

Outliers, mean and median

$\text{outlier} \Rightarrow \text{mean shifts, median barely moves}$

A single extreme value drags the mean toward it while the median stays near the middle. This idea appears on nearly every test.

Not given — geometry beyond the reference sheet

Equation of a circle

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

Center $(h, k)$ and radius $r$ . Complete the square in both variables to convert a scattered equation into this form.

Arc length

$s = \frac{\theta}{360^\circ} \times 2\pi r = r\theta_{\text{rad}}$

A fraction of the circumference set by the central angle. The radian version is cleaner when the angle is already in radians.

Sector area

$A = \frac{\theta}{360^\circ} \times \pi r^2 = \tfrac{1}{2} r^2 \theta_{\text{rad}}$

The same fraction of the whole circle area. Arc length and sector area always use the identical fraction.

Inscribed angle theorem

$\theta_{\text{inscribed}} = \tfrac{1}{2}\theta_{\text{central}}$

An angle drawn from the circle to an arc is half the central angle on the same arc. An angle inscribed in a semicircle is therefore a right angle.

Area of a parallelogram

$A = bh$

Base times perpendicular height, not the slanted side length. A rhombus is the case where all four sides are equal.

Area of a trapezoid

$A = \tfrac{1}{2}(b_1 + b_2)h$

Average of the parallel sides times the height. Not on the reference sheet despite appearing regularly.

Sum of interior angles of a polygon

$S = (n - 2) \times 180^\circ$

For $n$ sides. Each interior angle of a regular polygon is $S/n$ , and each exterior angle is $360^\circ/n$ .

Similar figures: length, area and volume

$k \;:\; k^2 \;:\; k^3$

If lengths scale by $k$ , areas scale by $k^2$ and volumes by $k^3$ . Applying $k$ to an area is a very common wrong answer.

Not given — trigonometry

SOHCAHTOA

$\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}$

Right triangles only. The SAT trigonometry questions are almost entirely these three ratios applied to a labeled diagram.

Cofunction identity

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

The sine of an angle equals the cosine of its complement. The SAT tests this relationship directly and often.

Tangent as a ratio

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

Useful when a question gives you two of the three ratios and asks for the third.

Pythagorean identity

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

Recovers one ratio from the other without finding the angle. Watch the sign when the quadrant matters.

Degree and radian conversion

$\text{radians} = \text{degrees} \times \frac{\pi}{180}$

The reference sheet states that a circle has $2\pi$ radians but not this conversion. Multiply by $180/\pi$ to go the other way.

Frequently asked questions

Yes. A reference sheet is available on every math question in the digital testing app. It contains fifteen geometry facts: the area and circumference of a circle, the area of a rectangle and a triangle, the Pythagorean theorem, the two special right triangles, five volume formulas, and the facts that a circle has $360^\circ$ or $2\pi$ radians and a triangle has $180^\circ$ . Nothing algebraic is included.

Everything outside geometry, and some geometry too. The essentials are slope and the equations of a line, the distance and midpoint formulas, the quadratic formula and vertex form, the discriminant, exponent rules, exponential growth and compound interest, percent change, the mean, probability, the equation of a circle, arc length and sector area, the interior angle sum, and SOHCAHTOA with the cofunction identity.

Yes. The PSAT/NMSQT and the digital SAT use the same reference sheet and the same four math content domains, so a single set of memorized formulas covers both. The PSAT is shorter and its question difficulty ceiling is slightly lower, but nothing on the formula side differs.

Slope, $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ , because Algebra is about 35% of the section and nearly every linear question, in graph or word-problem form, reduces to a rate of change. The quadratic formula is second, since Advanced Math is another 35% and it solves any quadratic when factoring fails.

SAT Math Reference Sheet Formulas