The Ultimate ACT Math Formula Cheat Sheet

Numbers, percents and rates

Mixed number

a + c/b = (ab + c)/b

b ≠ 0; keep the sign of a negative mixed number outside the entire positive quantity.

Percent of a whole

part = (percent/100) × whole

Write 12.5% as 0.125 before multiplying.

Percent change

(new − old)/old × 100%

Old value must be nonzero; a negative result indicates a decrease.

Proportion

a/b = c/d ⇒ ad = bc

b and d must be nonzero.

Distance and rate

distance = rate × time

Use matching units. Average speed is total distance/total time.

Simple interest

I = Prt

P is principal; r is a decimal annual rate; t is time in years.

Order of operations

Grouping → powers → × and ÷ → + and −

Work multiplication/division left to right; then addition/subtraction left to right.

Absolute value

|x| = x for x ≥ 0; |x| = − x for x < 0

Absolute value is nonnegative, including 0.

Algebra, exponents and functions

Distributive property

a(b + c) = ab + ac

Distribute to every term.

Difference of squares

a² − b² = (a − b)(a + b)

A sum of squares does not factor this way over the reals.

Perfect-square trinomials

a² ± 2ab + b² = (a ± b)²

Use matching signs.

Quadratic formula

x = [− b ± √(b² − 4ac)]/(2a)

For ax² + bx + c = 0, a ≠ 0. Divide the entire numerator by 2a.

Discriminant

D = b² − 4ac

For a quadratic with a ≠ 0: D >0 gives two real roots; D = 0: one repeated real root; D <0: no real roots.

Same-base powers

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ

For integer exponents; use a ≠ 0 for zero or negative exponents and for division.

Power of a power

(aᵐ)ⁿ = aᵐⁿ

For integer exponents; a must be nonzero if an exponent is zero or negative. Do not extend this rule to arbitrary radicals.

Zero and negative powers

a⁰ = 1; a⁻ⁿ = 1/aⁿ

a ≠ 0.

Square roots

√(ab) = √a × √b; √(a²) = |a|

For the product rule here, a and b are nonnegative.

Scientific notation

a × 10ⁿ, with 1 ≤ |a| <10

n is an integer; zero is handled separately.

Function notation

f(c) means substitute c for every input variable

If f(x) = x² − 3x, then f(4) = 16 − 12 = 4.

Logarithms

log_b(xy) = log_b x + log_b y

b >0, b ≠ 1, x >0 and y >0.

Log quotient and power

log_b(x/y) = log_b x − log_b y; log_b(xʳ) = r log_b x

Use the same positive-base and positive-input restrictions.

Change of base

log_b x = (log x)/(log b)

b >0, b ≠ 1, x >0; numerator and denominator use the same log base.

Coordinate geometry

Slope

m = (y₂ − y₁)/(x₂ − x₁)

x₂ ≠ x₁; a vertical line has undefined slope.

Line equations

y = mx + b; y − y₁ = m(x − x₁)

b is the y-intercept. A vertical line has equation x = constant.

Parallel and perpendicular lines

Parallel nonvertical lines: equal slopes. Perpendicular: m₁m₂ =− 1

Distinct vertical lines are parallel. The perpendicular product rule needs defined slopes; horizontal and vertical lines are also perpendicular.

Distance

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Subtract coordinates first, square each difference, add, then take the square root.

Midpoint

[(x₁ + x₂)/2, (y₁ + y₂)/2]

Average x-coordinates and y-coordinates separately.

Geometry and trigonometry

Rectangle and parallelogram

Rectangle A = lw; parallelogram A = bh

h is perpendicular height, not the slanted side.

Triangle and trapezoid

Triangle A = bh/2; trapezoid A =(b₁ + b₂)h/2

For the trapezoid, b₁ and b₂ are the parallel bases.

Circle

A = πr²; C = 2πr = πd

d = 2r. Keep π exact until the problem calls for rounding.

Arc and sector

Arc length =(θ/360°)2πr; sector area =(θ/360°)πr²

These versions use θ in degrees.

Angles

Triangle angle sum = 180°; polygon interior-angle sum =(n − 2)180°

For a simple n-sided polygon. An equilateral triangle has three 60° angles.

Right triangle

a² + b² = c²

c is the hypotenuse opposite the 90° angle.

Special right triangles

45°–45°–90°: x, x, x√2; 30°–60°–90°: x, x√3, 2x

Sides are listed opposite the smaller angles first; the last is the hypotenuse.

Right-triangle ratios

sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent

Opposite and adjacent depend on the chosen acute angle.

Trig identity and angle units

sin²θ + cos²θ = 1; 180° = π radians

Match calculator angle mode to the problem.

Law of sines

a/sin A = b/sin B = c/sin C

Each lowercase side is opposite its matching uppercase angle.

Law of cosines

c² = a² + b² − 2ab cos C

C is the angle between sides a and b.

Prism and cylinder volume

Prism V = Bh; cylinder V = πr²h

B is base area; h is perpendicular height.

Pyramid and cone volume

Pyramid V = Bh/3; cone V = πr²h/3

Use perpendicular height, not slant height.

Sphere

V = 4πr³/3; surface area = 4πr²

Volume uses cubic units; surface area uses square units.

Surface area

Rectangular prism SA = 2(lw + lh + wh); closed cylinder SA = 2πrh + 2πr²

Include both circular bases for a closed cylinder.

Cone surface area

SA = πrs + πr²

For a right circular cone including its base; s is slant height.

Data, probability and matrices

Mean and weighted mean

Mean = sum/count; weighted mean = Σ(wx)/Σw

Weights represent frequencies or relative contributions; total weight must be nonzero.

Median, mode and range

Median: middle of sorted data. Mode: most frequent value. Range: maximum − minimum.

For an even number of data values, average the two middle values. A data set can have multiple modes or no repeated value.

Equally likely outcomes

P(event) = favorable outcomes/total outcomes

Use this counting formula only when elementary outcomes are equally likely.

Complement and union

P(not A) = 1 − P(A); P(A or B) = P(A) + P(B) − P(A and B)

Subtract the overlap to avoid counting it twice.

Independent events

P(A and B) = P(A)P(B)

Use multiplication this way only when events are independent.

Factorial

n! = n(n − 1)…1; 0! = 1

n is a nonnegative integer.

2× 2 determinant

For rows [a, b] and [c, d], determinant = ad − bc

Multiply the main diagonal and subtract the other diagonal product.

Turn a formula into a useful skill

First identify what the question asks for. Label the known quantities, choose a relationship, and check units before substituting. After solving, ask whether the size and sign of the answer make sense.

Example: a circle has diameter 10. Its radius is 5, so its area is π × 5² = 25π square units. Using 10 as the radius would make the answer four times too large.

Example: the line through (2,3) and (6,11) has slope (11 − 3)/(6 − 2) = 2. Through the first point, y − 3 = 2(x − 2), which simplifies to y = 2x − 1. Both original points satisfy that equation.

Example: scores 70 and 90 with weights 1 and 3 have weighted mean (70 × 1 + 90 × 3)/(1 + 3) = 85. Averaging the two scores without their weights would give 80.

Use ACT Math worksheets to work on one skill, or revisit the rules of exponents, quadratic formula, law of sines and law of cosines lessons when a rule feels unfamiliar.

The ACT Math Online Center connects lessons, practice and book-selection options. Review a book’s actual preview and format before choosing paid material. Formula recall helps, but it does not replace interpreting graphs, modeling a situation or explaining a solution.

Calculators are allowed for Math under ACT’s calculator policy; the online test includes Desmos. Check your permitted model and angle mode before practicing. This review does not require a purchase or email signup.