The Digital SAT Math Formula Sheet

Every formula for the digital SAT in one place, the ones Bluebook gives you, the ones you have to memorize, and the part that actually moves scores: how each one gets tested. Free, from the tutors at High Performance Tutoring.

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On the test

What the SAT Gives You

These figures and formulas sit in the Bluebook reference panel on every math question. You don't memorize them, you just have to know when they apply and read the givens carefully.

Circle

\( A = \pi r^2 \)

\( C = 2\pi r \)

Rectangle

\( A = \ell w \)

Triangle

\( A = \tfrac{1}{2}bh \)

Right Triangle

\( a^2 + b^2 = c^2 \)

\( s \) \( s \) \( s\sqrt{2} \)

45-45-90

\( x \) \( x\sqrt{3} \) \( 2x \)

30-60-90

Box

\( V = \ell wh \)

Cylinder

\( V = \pi r^2 h \)

Sphere

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

Cone

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

Pyramid

\( V = \tfrac{1}{3}\ell wh \)

A full circle is \( 360^\circ \) and \( 2\pi \) radians · a triangle's angles sum to \( 180^\circ \).

The catch: the sheet gives you the cone's volume, but not the reminder that the SAT loves to hand you the volume and ask for the radius (run the formula backward). And the single most common geometry mistake isn't on the sheet at all: using the diameter as the radius. Check which one you were given, every time.

Memorize these

What's Not on the Sheet

This is where most points are won and lost. None of these appear in the reference panel, know them cold.

Algebra

Lines & Slope

Slope-intercept form

\( y = mx + b \)

\( m \) = slope
\( b \) = y-intercept

Point-slope form

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

\( m \) = slope
\( (x_1, y_1) \) = a point on the line

Standard form

\( ax + by = c \)

Slope between two points

\( m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1} \)

Parallel lines

\( m_1 = m_2 \)

Perpendicular lines

\( m_1 \cdot m_2 = -1 \)

Perpendicular slopes are negative reciprocals, flip and negate.

Midpoint

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

Distance between two points

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

Systems of Equations

Standard form of a system

\[ \begin{cases} ax + by = c \\ dx + ey = f \end{cases} \]

Two linear equations, two unknowns. The solution is the \( (x, y) \) that satisfies both.

Elimination method

Scale both equations so one variable's coefficients are opposites.
Add the equations to cancel that variable.
Solve for the variable that remains.
Substitute that value into either equation to find the other variable.

Substitution method

Isolate one variable in one of the equations.
Substitute that expression into the other equation.
Solve for that single remaining variable.
Substitute back to find the other variable.

One solution

No solution

Infinitely many solutions

Inequalities

Solving inequalities

\( -2x < 6 \ \Rightarrow\ x > -3 \)

Solve like an equation, but flip the sign when you multiply or divide by a negative.

Systems of inequalities

The solution is the region where the shaded areas overlap.

Advanced Math

Quadratics

Quadratic equation

\( ax^2 + bx + c = 0 \)

Quadratic formula

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

Discriminant

\( D = b^2 - 4ac \)

\( D > 0 \) → two real solutions
\( D = 0 \) → one real solution
\( D < 0 \) → no real solutions

Vertex form

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

Vertex \( = (h, k) \)

Factored form

\( y = a(x - p)(x - q) \)

x-intercepts are \( x = p \) and \( x = q \)

Standard form

\( y = ax^2 + bx + c \)

y-intercept \( = c \)

x-value of the vertex

\( h = -\dfrac{b}{2a} \)

Sum & product of the roots

\( r + s = -\dfrac{b}{a} \qquad r \cdot s = \dfrac{c}{a} \)

where \( r \) and \( s \) are the solutions to \( ax^2 + bx + c = 0 \).

Polynomials & Factoring

Difference of squares

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

Perfect-square trinomials

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

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

Zero-product property

\( (x - p)(x - q) = 0 \ \Rightarrow\ x = p \text{ or } q \)

Factor, zero, and intercept

\[ \begin{gathered} (x - p) \text{ is a factor} \\ \Updownarrow \\ f(p) = 0 \\ \Updownarrow \\ (p,\,0) \text{ is an } x\text{-intercept} \end{gathered} \]

Three ways to say the same thing.

Functions

Function notation

\( f(x) \)

\( f(3) \) = the output when \( x = 3 \)

Transformations of \( f(x) \)

\( f(x) + c \) → shifts up
\( f(x) - c \) → shifts down
\( f(x + h) \) → shifts left
\( f(x - h) \) → shifts right
\( -f(x) \) → reflects over the x-axis
\( f(-x) \) → reflects over the y-axis

Composite functions

\( f(g(x)) \)

Work inside-out: \( g \) first, then \( f \)

Exponent Rules

\( x^a \cdot x^b = x^{a+b} \)

\( \dfrac{x^a}{x^b} = x^{a-b} \)

\( (x^a)^b = x^{ab} \)

\( (xy)^a = x^a y^a \)

\( \left(\dfrac{x}{y}\right)^a = \dfrac{x^a}{y^a} \)

\( x^{-a} = \dfrac{1}{x^a} \)

\( x^0 = 1 \)

\( x^{m/n} = \sqrt[n]{x^m} \)

Exponential Functions

Exponential model

\( y = a \cdot b^x \)

\( a \) = starting value
\( b > 1 \) → growth
\( 0 < b < 1 \) → decay

Growth / decay over time

\( y = a(1 + r)^t \)

\( y = a(1 - r)^t \)

\( +r \) growth, \( -r \) decay
\( r \) = rate as a decimal, \( t \) = time

Solving exponential equations

\( b^m = b^n \ \Rightarrow\ m = n \)

Rewrite both sides with the same base, then set the exponents equal.

Geometry & Trigonometry

Circles (Beyond the Sheet)

Equation of a circle

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

Arc length

\( L = \dfrac{\theta}{360} \cdot 2\pi r \)

Sector area

\( A = \dfrac{\theta}{360} \cdot \pi r^2 \)

Angles

Vertical angles

all red angles are equal
all green angles are equal
red and green angles are supplementary

Supplementary angles

Two angles that add to \( 180^\circ \).

Complementary angles

Two angles that add to \( 90^\circ \).

Parallel lines cut by a transversal

all red angles are equal
all green angles are equal
red and green angles are supplementary

Polygons

Triangle angles

\( A + B + C = 180^\circ \)

Polygon interior angles

\( \text{Sum} = (n - 2) \times 180^\circ \)

\( n \) = number of sides

Similar figures

\( k \) = scale factor

all corresponding angles are equal
sides scale by \( k \)
areas scale by \( k^2 \)
volumes scale by \( k^3 \)

Right-Triangle Trig

SOH-CAH-TOA

\( \sin\theta = \dfrac{\text{opp}}{\text{hyp}} \)

\( \cos\theta = \dfrac{\text{adj}}{\text{hyp}} \)

\( \tan\theta = \dfrac{\text{opp}}{\text{adj}} \)

Pythagorean triples

Complementary angles

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

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

Surface Area

Rectangular prism (box)

\( SA = 2(\ell w + \ell h + wh) \)

Cylinder

\( SA = 2\pi r^2 + 2\pi rh \)

Sphere

\( SA = 4\pi r^2 \)

Problem-Solving & Data Analysis

Rates, Percent & Proportions

Percent change

\( P = \dfrac{(\text{new}) - (\text{old})}{(\text{old})} \times 100\% \)

\( P > 0 \) → percentage increase
\( P < 0 \) → percentage decrease

Percent increase, decrease & of

\( \text{percentage increase: } (\text{new}) = (\text{old}) \times (1 + r) \)

\( \text{percentage decrease: } (\text{new}) = (\text{old}) \times (1 - r) \)

\( \text{percent of: } (\text{part}) = r \times (\text{whole}) \)

\( r \) = the percent written as a decimal (\( 20\% \to r = 0.20 \))

Distance, rate & time

\( (\text{distance}) = (\text{rate}) \times (\text{time}) \)

\( (\text{work}) = (\text{rate}) \times (\text{time}) \)

Direct variation

\( y = kx \)

Inverse variation

\( y = \dfrac{k}{x} \)

Statistics & Data

Mean (average)

\( (\text{mean}) = \dfrac{(\text{sum of values})}{(\text{number of values})} \)

\( (\text{sum of values}) = (\text{mean}) \times (\text{number of values}) \)

Median

The middle number in an ordered list.

Mode

The most frequently occurring value in a list.

Range (spread)

\( (\text{range}) = (\text{max}) - (\text{min}) \)

Probability

\( P(A) = \dfrac{(\#\text{ ways } A \text{ can happen})}{(\#\text{ total possible outcomes})} \)

The HPT difference

The Formulas Are Easy. The Traps Are the Test.

A formula list won't tell you how the SAT turns a formula you "know" into a wrong answer. These four cost more points than any missing formula.

1. The Swap

\( y = 6x + 4 \) vs \( y = 4x + 6 \), mixing up slope and intercept is the #1 linear-equation error. Say it before you submit: rate on the x, start alone.

2. The Double-Cross

A radius/diameter mix-up quietly multiplies your circle answer by 2, 4, or 8. The number they hand you is usually the one you don't want.

3. The Lost Twin

The \( \pm \) means two answers. Find both, then report the one actually asked for (positive? smallest? the one that fits the context?).

4. The Phantom Cancel

You can only cancel common factors, never terms. \( \dfrac{x+3}{x+5} \neq \dfrac{3}{5} \). Factor first.

Ready when you are

Get the Formulas and the Strategy

Print this and drill it until the memorize section is automatic. The score jump comes from the next step, knowing which formula a question is secretly asking for, and sidestepping the trap built around it. That's exactly what we teach, one-on-one.