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Exam prep · ACT · Unit guide

ACT: Math

From solving equations to right triangles, the core math skills explained clearly

On the exam: Pre-algebra through algebra II, coordinate and plane geometry, and a little trigonometry.

Take the ACT quiz Make flashcards from my notes

Think you can solve for x in isolation but freeze the moment it's hidden inside a word problem? That's where most students lose points on the ACT math section. In the next few minutes we'll move from basic algebra through coordinate geometry, plane geometry, and trigonometry, then solve a full question together.

Solving Linear Equations

Every algebra problem comes down to isolating a variable. Take three x plus five equals twenty. First you subtract five from both sides, leaving three x equals fifteen. Then you divide both sides by three, and x equals five. The rule is simple: whatever you do to one side of the equation, you must do to the other, keeping it balanced.

3x + 5 = 20
3x = 15
x = 5

Key points.

Isolate the variable step by step
Keep both sides of the equation balanced
Check your answer by substituting back

Factoring Quadratics

Quadratic equations show up constantly, written as x squared minus five x plus six equals zero. You usually want to factor rather than grind through the quadratic formula. Look for two numbers that multiply to six and add to negative five, which are negative two and negative three. That gives you x minus two times x minus three equals zero, so x equals two or x equals three.

x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x = 2 or x = 3

Key points.

Set the equation equal to zero
Find two numbers that multiply and add correctly
Write it as two binomials
Solve each factor for x

Coordinate Geometry Basics

On the coordinate plane, slope tells you how steep a line is: rise over run, or the change in y divided by the change in x. Distance between two points comes straight from the Pythagorean theorem, squaring the differences in x and y, adding them, and taking the square root. The midpoint is just the average of the two x values and the average of the two y values.

Key points.

Slope equals rise over run
Distance formula uses the Pythagorean theorem
Midpoint averages the x and y values
Line equation: y equals m x plus b

Triangles and Circles

Plane geometry leans heavily on two shapes. For triangles, the three interior angles always add up to 180 degrees, and for right triangles the Pythagorean theorem, a squared plus b squared equals c squared, connects the two legs to the hypotenuse. For circles, circumference is two pi r and area is pi r squared, and an arc length is just that fraction of the full circle you're measuring.

Triangles

Interior angles sum to 180 degrees
a² + b² = c² for right triangles
30-60-90 and 45-45-90 have fixed ratios

Circles

Circumference equals 2 pi r
Area equals pi r squared
Arc length is a fraction of circumference

Right Triangle Trigonometry

Trigonometry questions almost always reduce to SOH CAH TOA. Sine of an angle is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Picture a right triangle with legs three and four and hypotenuse five: the sine of the angle opposite the side of length three is three fifths, and the cosine of that same angle is four fifths.

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
legs 3, 4, hyp 5 → sin θ = 3/5

Key points.

SOH: sine is opposite over hypotenuse
CAH: cosine is adjacent over hypotenuse
TOA: tangent is opposite over adjacent

Worked Question

Let's work a typical question. A rectangle's length is three more than twice its width, and its perimeter is thirty six. What is the width? Call the width w, so the length is two w plus three. Perimeter equals two w plus two l, which equals thirty six. Substituting gives six w plus six equals thirty six, so w equals five.

let w = width, l = 2w + 3
2w + 2l = 36
2w + 2(2w+3) = 36
6w + 6 = 36
w = 5

Key points.

Translate the words into variables
Write the perimeter equation
Substitute and simplify
Solve for the unknown

Arithmetic vs Geometric Sequences

Tests like this one check whether you can tell these apart. An arithmetic sequence adds the same number every time, like two, five, eight, eleven, where you keep adding three. A geometric sequence multiplies by the same number every time, like two, six, eighteen, fifty four, where you keep multiplying by three. One grows in a straight line, the other grows exponentially, and mixing them up is a common mistake.

Arithmetic

Add a constant difference each term
Grows in a straight line
Formula: a(n) = a1 + (n−1)d

Geometric

Multiply by a constant ratio each term
Grows exponentially
Formula: a(n) = a1 × r^(n−1)

Working Through the Test

When you sit down with a math question, read it twice so you know exactly what's being asked. Translate the words into an equation or a quick sketch before you touch your calculator. Solve it, then check your answer against the choices given, since a wrong option often matches a common error. If you get stuck, try plugging the answer choices back into the problem instead.

Read the question twice before solving
Translate words into an equation or diagram
Solve, then check against the answer choices
Plug in numbers when you get stuck
Keep an eye on your pacing

Remember this

The one thing to remember: translate every word problem into an equation before you reach for a calculator. Review these formulas on your own, then quiz yourself with real ACT practice problems to build the speed this section rewards.

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