Every Formula You Need for ACT Math

The ACT math section gives you 60 questions in 60 minutes and no formula sheet. Everything you need has to come from memory. This guide covers every major formula across all five tested topic areas so you can walk in fully prepared.

1. Geometry

Roughly 35 to 40% of ACT math is geometry. These formulas cover area, perimeter, volume, and angle relationships.

Area of a Triangle

A = ½ × b × h

b is the base and h is the height drawn perpendicular to that base. This works for every triangle, not just right triangles.

Pythagorean Theorem

a² + b² = c²

Only applies to right triangles. c is always the hypotenuse, which is the longest side opposite the right angle. Memorize the common triples: 3-4-5, 5-12-13, and 8-15-17 and their multiples.

Special Right Triangles

30-60-90 triangle → sides in ratio 1 : √3 : 2 45-45-90 triangle → sides in ratio 1 : 1 : √2

These come up constantly. In a 30-60-90 triangle, if the short leg is x, the long leg is x√3 and the hypotenuse is 2x. In a 45-45-90 triangle, both legs are equal and the hypotenuse is x√2.

Circle — Area and Circumference

Area = πr² Circumference = 2πr = πd

r is radius and d is diameter. Remember that diameter = 2r. The ACT often gives you the diameter and expects you to halve it before plugging in.

Arc Length and Sector Area

Arc Length = (θ / 360) × 2πr Sector Area = (θ / 360) × πr²

θ is the central angle in degrees. Think of it as taking a fraction of the full circle. If the angle is 90°, you are taking one-quarter of the total arc or area.

Area of Quadrilaterals

Rectangle: A = length × width Trapezoid: A = ½ × (b₁ + b₂) × h Parallelogram: A = base × height

For the trapezoid, b₁ and b₂ are the two parallel sides and h is the perpendicular height between them. A square is just a rectangle where length = width, so A = s².

Volume Formulas

Rectangular box: V = l × w × h Cylinder: V = πr²h Sphere: V = (4/3)πr³ Cone: V = (1/3)πr²h

Notice that a cone is exactly one-third of a cylinder with the same base and height. That connection makes the cone formula much easier to remember. The sphere formula is the one most students need to commit to memory on its own.

Sum of Interior Angles of a Polygon

Sum = (n - 2) × 180°

n is the number of sides. Triangle = 180°, quadrilateral = 360°, pentagon = 540°, hexagon = 720°. Divide by n to find each interior angle of a regular polygon.

Study Tip — Geometry

When a problem gives you a figure, write all known measurements directly on it. Label side lengths, mark angle values, and draw in any missing lines you need like altitudes or diagonals. Most geometry problems become straightforward once everything is labeled clearly.

2. Algebra

Algebra and functions make up about 24% of the test. Know your lines, quadratics, and exponent rules inside and out.

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁)

Slope is rise over run. Subtract the y-values on top and the x-values on the bottom. It does not matter which point you call (x₁, y₁) as long as you are consistent throughout the problem.

Forms of a Linear Equation

Slope-intercept: y = mx + b Point-slope: y - y₁ = m(x - x₁) Standard form: Ax + By = C

Slope-intercept is most common on the ACT. Point-slope is useful when you have a point and a slope but no y-intercept. Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Solves ax² + bx + c = 0 for any quadratic. The expression under the square root is called the discriminant (b² - 4ac). If it is positive you get two real solutions, zero gives one solution, and negative means no real solutions.

Factoring Patterns

Difference of squares: a² - b² = (a + b)(a - b) Perfect square: (a + b)² = a² + 2ab + b² Perfect square: (a - b)² = a² - 2ab + b²

Recognizing difference of squares instantly saves time. When you see x² - 25, factor it to (x + 5)(x - 5) right away. The perfect square patterns let you expand binomials quickly without multiplying every term out by hand.

Distance Formula

d = √((x₂ - x₁)² + (y₂ - y₁)²)

This is the Pythagorean theorem applied to a coordinate plane. The horizontal and vertical distances between two points form the two legs of a right triangle, and you are solving for the hypotenuse.

Midpoint Formula

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

You are just averaging the x-coordinates and averaging the y-coordinates of the two points. The result is the point exactly halfway between them.

Exponent Rules

aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ a⁻ⁿ = 1 / aⁿ a^(1/n) = nth root of a

Negative exponents mean move the base to the denominator. Fractional exponents mean roots: x^(1/2) is the square root and x^(1/3) is the cube root. These rules appear in nearly every algebra and pre-calc problem.

Vertex of a Parabola

x-coordinate of vertex = -b / 2a

For y = ax² + bx + c, plug -b/2a back into the equation to get the y-coordinate of the vertex. The vertex is the maximum point if a is negative and the minimum point if a is positive.

Study Tip — Algebra

When a question asks which expression is equivalent to something, try plugging in a simple number like 2 or 3 for the variable. Evaluate the original expression and each answer choice to see which one matches. This is often faster than pure algebra and avoids careless sign errors.

3. Trigonometry

Trig makes up about 7% of the test. Know your basic ratios, the Pythagorean identity, and the most common angle values.

SOH-CAH-TOA

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

All three ratios refer to a right triangle with respect to a specific angle. SOH-CAH-TOA is the standard memory device. Also know the reciprocal functions: csc = 1/sin, sec = 1/cos, and cot = 1/tan.

Pythagorean Identity

sin²θ + cos²θ = 1 tan θ = sin θ / cos θ

The most important trig identity on the ACT. Rearranged: cos²θ = 1 - sin²θ and sin²θ = 1 - cos²θ. Use these to swap between sine and cosine when simplifying expressions on the test.

Common Angle Values

Angle sin cos tan 0° 0 1 0 30° 1/2 √3/2 1/√3 45° √2/2 √2/2 1 60° √3/2 1/2 √3 90° 1 0 undefined

Memorize the sine pattern: 0, 1/2, √2/2, √3/2, 1. Cosine runs the same values in reverse. Knowing these cold saves a lot of time on trig questions.

Law of Sines

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

Works for any triangle, not just right triangles. Use it when you know two angles and a side, or two sides and a non-included angle.

Law of Cosines

c² = a² + b² - 2ab · cos C

Use when you know two sides and the included angle between them. It is a generalization of the Pythagorean theorem. When C = 90°, the last term disappears and you are left with a² + b² = c².

4. Statistics and Probability

Stats and probability appear in data interpretation problems and calculation questions throughout the test.

Mean (Average)

Mean = Sum of all values / Count of values Sum = Mean × Count

Add everything up and divide by how many values there are. The ACT often gives you the mean and the count and asks for the total sum. Rearranging to Sum = Mean × Count is the key move for those problems.

Probability

P(event) = favorable outcomes / total outcomes P(A or B) = P(A) + P(B) - P(A and B) P(A and B) = P(A) × P(B) [if independent]

For independent events, multiply the probabilities. For either/or, add them and subtract any overlap. If the two events cannot both happen at the same time (mutually exclusive), there is no overlap so you just add them directly.

Combinations and Permutations

Combinations: nCr = n! / (r! × (n - r)!) Permutations: nPr = n! / (n - r)!

Use combinations when order does not matter (choosing committee members). Use permutations when order does matter (awarding first, second, and third place). Most calculators you bring to the ACT have nCr and nPr buttons built in.

Median, Mode, and Range

Median = middle value when data is sorted Mode = value that appears most often Range = maximum value - minimum value

Always sort the data first before finding the median. For an even number of values, average the two middle ones. These show up frequently in data reading and interpretation problems.

5. Numbers and Pre-Calc

These topics show up in the harder questions near the end of the test. Thorough prep here is where students pick up extra points.

Arithmetic Sequence

aₙ = a₁ + (n - 1) × d

a₁ is the first term, d is the common difference (what you add each step), and n is the position of the term you want. Example: 5, 9, 13, 17 ... has d = 4, so the 10th term is 5 + (9)(4) = 41.

Geometric Sequence

aₙ = a₁ × r^(n - 1)

r is the common ratio (what you multiply each step). Example: 3, 6, 12, 24 ... has r = 2, so the 6th term is 3 × 2⁵ = 96. If r is between negative 1 and 1, the terms shrink toward zero.

Logarithm Rules

log(a × b) = log a + log b log(a / b) = log a - log b log(aⁿ) = n × log a logₐ(a) = 1

Remember that log base b of x = y means b raised to the power y equals x. Logs and exponents are inverses of each other. If you see log₂(8), you are asking what power of 2 gives 8, and the answer is 3 because 2³ = 8.

Percent Change

% change = ((new value - old value) / old value) × 100

A positive result means an increase and a negative result means a decrease. The original value (old) always goes in the denominator. Do not confuse the percent change with the actual change in value.

Simple and Compound Interest

Simple: A = P(1 + rt) Compound: A = P(1 + r/n)^(nt)

P = principal, r = annual interest rate as a decimal, t = years, and n = number of times per year interest compounds. The ACT usually sticks to simple interest or annual compounding.

Absolute Value Equations and Inequalities

|x| = a → x = a or x = -a |x| < a → -a < x < a |x| > a → x < -a or x > a

Absolute value equations always produce two cases because the expression inside can be positive or negative. For inequalities, think of it as distance from zero on a number line. The ACT tests this in both equation and inequality form.

How to Get the Most Out of This Guide

Reading formulas is step one. Write each one on a flashcard and practice active recall every day for two weeks. Then do full timed practice sections under real test conditions. The goal is for these formulas to feel completely automatic so your mental energy on test day goes toward solving problems, not remembering equations.