SAT Math

SAT Math formulas

The formulas SAT Math questions use most, on one page and grouped by topic. The SAT gives you a reference sheet with some geometry formulas, but not the algebra, percent, or statistics ones, so those are worth knowing by heart.

Each group links to a concept guide with worked SAT-style examples (not official College Board questions).

Algebra

Slope-intercept form

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Slope-intercept form

$y = mx + b$

$m$ is the slope, $b$ is the $y$ -intercept. Solve for $y$ to get here.

Slope from two points

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

Rise over run. Subtract in the same order on the top and the bottom.

Point-slope form

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

Useful when you know the slope and one point. Distribute and solve for $y$ to reach slope-intercept form.

Parallel and perpendicular

$m_2 = m_1 \quad\text{or}\quad m_1 \cdot m_2 = -1$

Parallel lines share a slope. Perpendicular slopes are negative reciprocals, like 2 and $-\tfrac{1}{2}$ .

Interval notation

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Square bracket [ ]

$\le \text{ or } \ge \;\;\to\;\; [\;\;]$

The endpoint is part of the set. On a number line, it is a filled-in dot.

Parenthesis ( )

$< \text{ or } > \;\;\to\;\; (\;\;)$

The endpoint is not part of the set. On a number line, it is an open dot.

Infinity

$(-\infty, a] \quad\text{or}\quad [a, \infty)$

$\infty$ and $-\infty$ always get a parenthesis.

Union for “or”

$(-\infty, -2) \cup [3, \infty)$

Two separate pieces join with $\cup$ . This one means $x < -2$ or $x \ge 3$ .

Advanced Math

The quadratic formula

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Quadratic formula

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

Solves any quadratic. Set the equation equal to 0 first.

Discriminant

$D = b^2 - 4ac$

Positive: two solutions. Zero: one. Negative: none.

Vertex

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

The vertex's $x$ -coordinate, and the axis of symmetry. Plug it in to get $y$ .

Sum and product of roots

$r_1 + r_2 = -\dfrac{b}{a} \qquad r_1 r_2 = \dfrac{c}{a}$

Use these when a question asks for the sum or product, not the solutions.

Exponential growth and decay

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Exponential growth

$y = a(1 + r)^x$

$a$ is the start. 6% growth is a base of 1.06.

Exponential decay

$y = a(1 - r)^x$

A 20% decrease each step is a base of 0.8. The amount shrinks but never reaches zero.

Doubling or halving

$y = a \cdot 2^{t/d} \quad\text{or}\quad y = a\left(\tfrac{1}{2}\right)^{t/h}$

$d$ is the doubling time and $h$ is the half-life. After one $d$ , the amount has doubled.

Linear, for comparison

$y = mx + b$

Linear models add the same amount each step. Exponential models multiply by the same factor.

Problem-Solving and Data Analysis

Percent change

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Percent change

$\dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%$

The denominator is always the starting amount.

Increase by p%

$\text{new} = \text{original} \times \left(1 + \tfrac{p}{100}\right)$

A 15% raise is $\times 1.15$ . A 100% increase doubles the amount.

Decrease by p%

$\text{new} = \text{original} \times \left(1 - \tfrac{p}{100}\right)$

A 30% discount is $\times 0.70$ : you pay 70% of the price.

Finding the original

$\text{original} = \text{new} \div \text{multiplier}$

If a price after a 20% discount is $48, the original was $48 \div 0.8$ = $60.

Mean and median

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Mean

$\text{mean} = \dfrac{\text{sum of values}}{n}$

$n$ is the number of values. Every value affects the mean, including outliers.

Median

$\text{middle value, in order}$

With an even count, average the two middle values.

Sum from the mean

$\text{sum} = \text{mean} \times n$

The key step for questions that add or remove a value and ask for the new mean.

Range

$\text{range} = \max - \min$

A simple measure of spread. Standard deviation measures how far values typically sit from the mean.

Geometry and Trigonometry

The Pythagorean theorem

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Pythagorean theorem

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

$c$ is always the hypotenuse, the side across from the right angle.

Finding a leg

$a = \sqrt{c^2 - b^2}$

Subtract, do not add, when the missing side is a leg.

45-45-90 triangle

$x,\ x,\ x\sqrt{2}$

An isosceles right triangle. The hypotenuse is $\sqrt{2}$ times a leg. On the reference sheet.

30-60-90 triangle

$x,\ x\sqrt{3},\ 2x$

The short leg is across from $30^\circ$ , and the hypotenuse is twice it. On the reference sheet.

Volume of a cone and a pyramid

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Cone

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

$r$ is the base radius. $h$ is the straight-up height.

Pyramid

$V = \tfrac{1}{3}Bh$

$B$ is the base area. For a rectangle, $V = \tfrac{1}{3}lwh$ .

Cylinder, for comparison

$V = \pi r^2 h$

Same base and height, no $\tfrac{1}{3}$ . A cone is one third of it.

Height from slant height

$h = \sqrt{\ell^2 - r^2}$

Height, radius, and slant height form a right triangle.

Surface area of a sphere

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Sphere surface area

$SA = 4\pi r^2$

$r$ is the radius. The answer is in square units.

Hemisphere, total surface area

$SA = 3\pi r^2$

Half the sphere's curved surface ( $2\pi r^2$ ) plus the flat circular base ( $\pi r^2$ ).

Radius from surface area

$r = \sqrt{\dfrac{SA}{4\pi}}$

Divide the surface area by $4\pi$ , then take the square root.

Sphere volume, for comparison

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

On the reference sheet. Volume uses $r^3$ ; surface area uses $r^2$ .

Equation of a circle

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Standard form

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

Center $(h, k)$ , radius $r$ . A center at the origin gives $x^2 + y^2 = r^2$ .

Expanded form

$x^2 + y^2 + Dx + Ey + F = 0$

The same circle multiplied out. Complete the square to read it.

Completing the square

$x^2 + Dx = \left(x + \tfrac{D}{2}\right)^2 - \left(\tfrac{D}{2}\right)^2$

Halve the $x$ coefficient, square it, and add it to both sides. Do the same for $y$ .

Circle facts, for reference

$C = 2\pi r \qquad A = \pi r^2$

Both on the reference sheet. A tangent is perpendicular to the radius.

The unit circle

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A point on the circle

$(x, y) = (\cos\theta, \sin\theta)$

Cosine is how far across, sine is how far up.

Pythagorean identity

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

Every point on the circle satisfies $x^2 + y^2 = 1$ , so this holds for every angle.

Degrees to radians

$180^\circ = \pi \text{ radians}$

Multiply degrees by $\tfrac{\pi}{180}$ to get radians. Multiply radians by $\tfrac{180}{\pi}$ to get degrees.

Arc length

$s = r\theta$

With $\theta$ in radians. On the unit circle $r = 1$ , so the arc length equals the angle.

A learner having an idea, surrounded by maps and equations

Practice it out loud

Know the formula, then use it

Pick a formula and work an SAT-style question by voice. Aristotle asks guiding questions instead of giving answers, so you can do the next one on your own.

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