SAT math formulas: what's on the reference sheet and what to memorize

On every Math question, Bluebook gives you a reference sheet with basic geometry: area and circumference of a circle, area of a rectangle and a triangle, the Pythagorean theorem, the two special right triangles, five volume formulas, and three angle facts. Everything else you need to know yourself, including slope, the quadratic formula, exponent rules, percent change, and the equation of a circle. Since algebra and advanced math make up most of the section, the formulas you memorize matter more than the ones you're given.

What's on the reference sheet

The sheet opens from a button at the top of the screen during the Math section.

Shape or fact	Formula
Circle	$A = \pi r^2$ and $C = 2\pi r$
Rectangle	$A = \ell w$
Triangle	$A = \frac{1}{2}bh$
Pythagorean theorem	$c^2 = a^2 + b^2$
30-60-90 triangle	Sides $x$ , $x\sqrt{3}$ , and $2x$
45-45-90 triangle	Sides $s$ , $s$ , and $s\sqrt{2}$
Rectangular prism	$V = \ell wh$
Cylinder	$V = \pi r^2 h$
Sphere	$V = \frac{4}{3}\pi r^3$
Cone	$V = \frac{1}{3}\pi r^2 h$
Rectangular pyramid	$V = \frac{1}{3}\ell wh$

It also states that a circle has 360 degrees of arc, that a circle has $2\pi$ radians of arc, and that the angles of a triangle add up to 180 degrees.

You don't need to memorize these, but know them well enough that you don't lose time opening the sheet for the area of a triangle. Save the sheet for the volume formulas and the special right triangles.

Formulas to memorize: lines and linear equations

Algebra is about 35% of the Math section, according to College Board, and most of it is about lines.

Concept	Formula
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form	$y = mx + b$ , where $b$ is the $y$ -intercept
Point-slope form	$y - y_1 = m(x - x_1)$
Standard form	$Ax + By = C$ , with slope $-\frac{A}{B}$
Parallel lines	Equal slopes
Perpendicular lines	Slopes multiply to $-1$

For a system of two linear equations, you also need to know how to count solutions:

One solution: the slopes are different.
No solution: the slopes are equal and the intercepts are different, so the lines are parallel.
Infinitely many solutions: the equations describe the same line.

Formulas to memorize: quadratics and polynomials

Advanced Math is another 35% of the section. Quadratics show up in many forms, and each form tells you something different.

Form	What it shows
$y = ax^2 + bx + c$	The $y$ -intercept is $c$ . The vertex's $x$ -coordinate is $-\frac{b}{2a}$ .
$y = a(x - h)^2 + k$	The vertex is $(h, k)$ .
$y = a(x - r)(x - s)$	The $x$ -intercepts are $r$ and $s$ .

The quadratic formula solves any $ax^2 + bx + c = 0$ :

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

The discriminant, $b^2 - 4ac$ , tells you how many real solutions there are. It's positive for two, zero for one, and negative for none. The SAT likes to ask for the value of a constant that gives exactly one solution, so this one is worth knowing cold.

Two more facts about $ax^2 + bx + c = 0$ save time: the solutions add up to $-\frac{b}{a}$ and multiply to $\frac{c}{a}$ .

Know these factoring patterns on sight:

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

Formulas to memorize: exponents and exponential models

Rule	Formula
Product	$x^a \cdot x^b = x^{a+b}$
Quotient	$\frac{x^a}{x^b} = x^{a-b}$
Power	$(x^a)^b = x^{ab}$
Negative exponent	$x^{-a} = \frac{1}{x^a}$
Fractional exponent	$x^{\frac{m}{n}} = \sqrt[n]{x^m}$

Exponential growth and decay use $y = a(1 + r)^t$ and $y = a(1 - r)^t$ , where $a$ is the starting amount and $r$ is the rate as a decimal. A quantity that grows by 6% a year is multiplied by 1.06 each year. One that loses 6% is multiplied by 0.94.

Formulas to memorize: percentages, statistics, and probability

Percent change: $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$
Increase by $p\%$ : multiply by $1 + \frac{p}{100}$ . Decrease by $p\%$ : multiply by $1 - \frac{p}{100}$ .
Mean: the sum of the values divided by the number of values.
Median: the middle value when the data is in order, or the average of the two middle values.
Probability: the number of favorable outcomes divided by the total number of outcomes. For conditional probability from a table, divide by the total of the row or column the question limits you to.

You won't need to calculate a standard deviation. You will need to know that it measures spread, so a data set clustered close to its mean has a smaller standard deviation than one that's spread out.

Formulas to memorize: circles and trigonometry

The reference sheet stops at area and circumference, but circle questions go further.

Circle equation: $(x - h)^2 + (y - k)^2 = r^2$ , with center $(h, k)$ and radius $r$
Arc length: $\frac{\theta}{360} \cdot 2\pi r$ , with $\theta$ in degrees
Sector area: $\frac{\theta}{360} \cdot \pi r^2$
Degrees to radians: $180^\circ = \pi$ radians

The SAT Owl guide to circle questions covers completing the square to find the center and radius from an expanded equation.

For right triangles, you need SOH CAH TOA:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Also know that $\sin x^\circ = \cos(90^\circ - x^\circ)$ . This comes up often, because the two acute angles of a right triangle add up to 90 degrees. The right triangles and trigonometry guide has worked examples.

Other geometry facts worth knowing

Vertical angles are equal, and angles on a straight line add up to 180 degrees.
When a line crosses two parallel lines, corresponding and alternate interior angles are equal.
Similar triangles have proportional sides. If the sides scale by $k$ , area scales by $k^2$ and volume by $k^3$ .
The interior angles of a polygon with $n$ sides add up to $(n - 2) \cdot 180^\circ$ .
Distance between two points: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , which is the Pythagorean theorem on a grid.

Where Desmos helps and where it doesn't

The Desmos graphing calculator in Bluebook can find a parabola's vertex and zeros, the intersection of two graphs, and the solutions to an equation. That's a fast check on algebra. But many questions ask what a number means, such as which part of $y = 3(1.05)^t$ is the growth rate, or which form of a quadratic shows its vertex. Desmos can't answer those for you. You still need the formulas.

How to memorize them

Write the list from memory once a day for a week, then check it against this page. Anything you missed goes on a short list.
Learn each formula with a question. A formula you've used on five practice questions sticks better than one you've read 20 times.
Mix your practice. Random practice sets force you to recognize which formula a question needs, which is the real skill on test day. The free question bank is sorted by skill if you want to target one area first.

To find out which math skills cost you the most points, take SAT Owl's free 24-question diagnostic.