ACT Math

ACT Math Formula Sheet — No Formula Sheet Provided, Memorize These

Unlike the SAT, the ACT provides no reference formulas. Everything here must be committed to memory before test day. Covers geometry, algebra, trigonometry, statistics, logarithms, and matrices.

60 questions · 60 minutes · No formula sheet · All topics must be memorized

Take a Free ACT Practice Exam →ACT Math Guide ACT Top 100 Vocabulary

Important: The ACT Math section provides absolutely no formula sheet. Every formula you see here must be memorized. The ACT Math section tests trigonometry, logarithms, and matrices — topics the SAT does not include. Budget additional time to learn those sections if they are new to you.

Geometry

MEMORIZE

Area of a circle

A = πr²

r = radius

Circumference

C = 2πr = πd

d = diameter

Area of a rectangle

A = lw

l = length, w = width

Area of a triangle

A = ½bh

b = base, h = perpendicular height

Area of a trapezoid

A = ½(b₁ + b₂)h

b₁, b₂ = parallel sides; h = height

Area of a parallelogram

A = bh

b = base, h = height

Pythagorean theorem

a² + b² = c²

c = hypotenuse

30-60-90 triangle

sides: x : x√3 : 2x

Opposite: 30° : 60° : 90°

45-45-90 triangle

sides: x : x : x√2

Opposite: 45° : 45° : 90°

Volume of rectangular prism

V = lwh

l = length, w = width, h = height

Volume of cylinder

V = πr²h

r = base radius, h = height

Volume of sphere

V = (4/3)πr³

r = radius

Volume of cone

V = (1/3)πr²h

r = base radius, h = height

Surface area of sphere

SA = 4πr²

r = radius

Sum of interior angles

S = (n − 2) × 180°

n = number of sides of polygon

Algebra

MEMORIZE

Slope formula

m = (y₂ − y₁) / (x₂ − x₁)

Rise divided by run

Slope-intercept form

y = mx + b

m = slope, b = y-intercept

Distance formula

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Derived from Pythagorean theorem

Midpoint formula

M = ((x₁+x₂)/2 , (y₁+y₂)/2)

Average of x and y coordinates

Quadratic formula

x = [−b ± √(b² − 4ac)] / 2a

Solves ax² + bx + c = 0

Vertex form

y = a(x − h)² + k

Vertex at (h, k)

Discriminant

Δ = b² − 4ac

Δ > 0: two real roots; Δ = 0: one; Δ < 0: none

Difference of squares

a² − b² = (a + b)(a − b)

Key factoring identity

Perfect square trinomials

(a ± b)² = a² ± 2ab + b²

Both forms needed

Exponential growth/decay

y = a(1 ± r)ᵗ

a = initial, r = rate, t = time

Trigonometry

MEMORIZE

SOH-CAH-TOA

sin = opp/hyp · cos = adj/hyp · tan = opp/adj

For a right triangle

Pythagorean identity

sin²θ + cos²θ = 1

Derived from Pythagorean theorem

Tangent identity

tan θ = sin θ / cos θ

Fundamental trig identity

Reciprocal: cosecant

csc θ = 1 / sin θ

Hypotenuse over opposite

Reciprocal: secant

sec θ = 1 / cos θ

Hypotenuse over adjacent

Reciprocal: cotangent

cot θ = 1 / tan θ = cos θ / sin θ

Adjacent over opposite

Law of Sines

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

Use for non-right triangles with AAS or ASA

Law of Cosines

c² = a² + b² − 2ab · cos C

Use for SAS or SSS triangles

Arc length (radians)

s = rθ

r = radius, θ = angle in radians

Area of triangle (trig)

A = ½ab · sin C

Use when two sides and included angle are known

Degrees to radians

radians = degrees × π / 180

1 full circle = 2π radians = 360°

Unit circle key values

sin 30° = ½ · sin 45° = √2/2 · sin 60° = √3/2

Memorize the first quadrant; use symmetry for others

Statistics & Probability

MEMORIZE

Mean

x̄ = Σx / n

Sum of all values divided by number of values

Median

Middle value when sorted

Average of two middle values if n is even

Range

Range = max − min

A measure of spread

Probability

P(A) = favorable / total outcomes

Assumes equally likely outcomes

P(A or B)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Inclusion-exclusion principle

P(A and B) — independent

P(A ∩ B) = P(A) × P(B)

Only when A and B are independent

Permutations

P(n, r) = n! / (n − r)!

Ordered arrangements of r items from n

Combinations

C(n, r) = n! / [r!(n − r)!]

Unordered selections of r items from n

Percent change

% change = (new − old) / old × 100

Positive = increase, negative = decrease

Weighted average

x̄ = Σ(wᵢxᵢ) / Σwᵢ

Each value multiplied by its weight

Logarithms & Matrices

MEMORIZE

Product rule

log(ab) = log a + log b

Applies to any base

Quotient rule

log(a/b) = log a − log b

Applies to any base

Power rule

log(aⁿ) = n · log a

Applies to any base

Change of base

logₐ b = log b / log a

Convert between bases using any common log

Log and exponent inverse

logₐ(aˣ) = x and aˡᵒᵍₐˣ = x

They undo each other

2×2 matrix determinant

|a b; c d| = ad − bc

det of [[a, b], [c, d]]

Matrix addition

[a b; c d] + [e f; g h] = [a+e b+f; c+g d+h]

Add corresponding entries

Number & Percent

MEMORIZE

Simple interest

I = Prt

P = principal, r = annual rate, t = time in years

Compound interest

A = P(1 + r/n)^(nt)

n = compounds per year

Percent formula

Part = (Percent / 100) × Whole

Rearrange to find any of the three

Direct variation

y = kx

k = constant of variation

Inverse variation

y = k / x

As x increases, y decreases proportionally

ACT Math section overview

Questions

1 to 60, increasing difficulty

60 min

Time

1 minute per question average

Allowed

Calculator

All 60 questions

~7%

Trig share

About 4 trig questions per exam

~35%

Geometry share

Largest single topic group

1–36

Score range

Integer score per section

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