ACT Math Formulas to Memorize for ACT

Why This Matters

The ACT Math section tests whether you understand when and why to use each formula, not just whether you can plug in numbers. You'll face 60 questions in 60 minutes, so these formulas need to be automatic. The test also loves to combine concepts: a question might require you to find a circle's area, use that result in a volume calculation, and then interpret the answer in a real-world context.

The formulas below cluster around core mathematical principles: measuring flat shapes, measuring 3D objects, navigating coordinate planes, solving equations, and working with triangles. Don't just memorize each formula in isolation. Know what category it belongs to, when it applies, and how it connects to related formulas. That conceptual understanding is what separates a good score from a great one.

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Two-Dimensional Area Formulas

These formulas measure the space inside flat shapes. The key principle: area always involves multiplying two dimensions together, which is why area units are always squared.

Area of a Rectangle

• $A = lw$ — multiply length by width to find the space inside any rectangle or square
• Units are always squared — square feet, square meters, etc., because you're multiplying two linear measurements
• Foundation for other formulas — rectangles are the building blocks; triangle area is half a rectangle, and prism volume extends this into 3D

Area of a Triangle

• $A = \frac{1}{2}bh$ — half the base times the height, where height must be perpendicular to the base
• Works for ALL triangles — not just right triangles; the height might fall outside the triangle in obtuse cases
• Common ACT trap — the test often gives you side lengths instead of height, requiring you to calculate height first (often using the Pythagorean theorem or special right triangle ratios)

Area of a Circle

• $A = \pi r^2$ — square the radius first, then multiply by $\pi$
• Radius, not diameter — if given diameter, divide by 2 before using this formula
• Sector problems — for partial circles, multiply this area by the fraction of the circle (central angle divided by 360°)

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Three-Dimensional Volume and Surface Area

Volume measures space inside 3D objects (cubic units), while surface area measures the outside skin (square units). The pattern: volume = base area × height.

Volume of a Rectangular Prism

• $V = lwh$ — length times width times height; rectangle area extended into 3D
• Units are cubed — cubic feet, cubic meters, etc., because you're multiplying three dimensions
• Box problems — any question about boxes, rooms, or tanks likely needs this formula

Volume of a Cylinder

• $V = \pi r^2 h$ — circle area ($\pi r^2$) times height; same pattern as the prism
• Stacked circles concept — visualize a cylinder as circles stacked on top of each other
• Common applications — pipes, cans, tanks; the ACT loves practical cylinder problems

Surface Area of a Rectangular Prism

• $SA = 2(lw + lh + wh)$ — add the three different face areas, then double (since opposite faces match)
• Six faces total — two of each type: top/bottom, front/back, left/right
• Wrapping and painting problems — surface area tells you how much material covers the outside

Surface Area of a Cylinder

• $SA = 2\pi rh + 2\pi r^2$ — lateral surface (the curved part) plus two circular bases
• Label trick — the lateral area $2\pi rh$ is what you'd get if you unrolled the curved part into a flat rectangle; its width is the circumference ($2\pi r$) and its height is $h$
• Partial cylinders — if a cylinder is open (no top), subtract one $\pi r^2$ from the formula

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Circle Measurements

Circles have their own special formulas because they involve $\pi$. The key relationship: everything connects back to the radius.

Circumference of a Circle

• $C = 2\pi r$ or equivalently $C = \pi d$ — the distance around the circle
• Diameter = 2r — use whichever form matches what the problem gives you
• Arc length — for partial circles, multiply circumference by the fraction (central angle ÷ 360°)

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Coordinate Geometry Formulas

These formulas let you work with points and lines on the coordinate plane. They're all derived from the relationship between horizontal and vertical distances.

Slope Formula

• $m = \frac{y_2 - y_1}{x_2 - x_1}$ — "rise over run," the change in $y$ divided by change in $x$
• Positive slope = line goes up (left to right); negative slope = line goes down; zero slope = horizontal; undefined slope = vertical
• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (flip the fraction and change the sign, so a slope of $\frac{2}{3}$ becomes $-\frac{3}{2}$)

Slope-Intercept Form

• $y = mx + b$ — where $m$ is the slope and $b$ is the y-intercept (the point where the line crosses the y-axis)
• Reading a graph — if a question gives you a graph, identify where the line crosses the y-axis for $b$, then count rise/run for $m$
• Writing equations — once you have slope and a point, plug them into this form to build the full equation

Midpoint Formula

• $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ — average the x-coordinates and average the y-coordinates
• Center point applications — finding the center of a line segment or the middle of a diameter
• Works in reverse — if given the midpoint and one endpoint, you can solve for the other endpoint

Distance Formula

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ — derived directly from the Pythagorean theorem
• Creates a right triangle — the horizontal and vertical distances are legs; the direct distance is the hypotenuse
• Always square before adding — a common error is forgetting to square the differences

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Triangle Relationships

Triangles appear constantly on the ACT. These formulas and ratios help you find missing sides and angles quickly.

Pythagorean Theorem

• $a^2 + b^2 = c^2$ — for right triangles only; $c$ is always the hypotenuse (longest side, opposite the right angle)
• Finding any side — rearrange to find a leg: $a = \sqrt{c^2 - b^2}$
• Common Pythagorean triples — memorize 3-4-5, 5-12-13, and 8-15-17 (plus their multiples like 6-8-10). Recognizing these saves real time on the test.
• Testing for right triangles — if the equation holds true for three given side lengths, the triangle is a right triangle

SOHCAHTOA (Trigonometric Ratios)

These apply only to right triangles. The angle $\theta$ is always one of the two non-right angles.

• SOH: $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ — the side across from the angle divided by the longest side
• CAH: $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ — the side next to the angle (not the hypotenuse) divided by the longest side
• TOA: $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$ — useful when the hypotenuse isn't involved in the problem

Special Right Triangle Ratios

• 45-45-90 triangle: sides in ratio $1:1:\sqrt{2}$ — the two legs are equal; hypotenuse is leg × $\sqrt{2}$
• 30-60-90 triangle: sides in ratio $1:\sqrt{3}:2$ — shortest side opposite 30°, medium side ($\sqrt{3}$) opposite 60°, longest side (2) opposite 90°
• Time-saver — memorizing these lets you skip calculations entirely when you recognize these triangles

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Algebraic Problem-Solving

When geometry formulas won't help, these algebraic tools solve equations and find unknown values.

Quadratic Formula

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — solves any equation in the form $ax^2 + bx + c = 0$
• The discriminant ($b^2 - 4ac$) tells you the nature of solutions: positive = two real solutions, zero = one repeated solution, negative = no real solutions
• When to use it — when factoring isn't obvious or possible; the formula always works

Percent Change

• $\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$
• Always divide by the original value, not the new one. This trips up a lot of students.
• Works for both increases and decreases. A positive result means increase; negative means decrease.

Probability

• $P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$
• Probability is always between 0 and 1 (or 0% and 100%)
• For the probability of two independent events both happening, multiply their individual probabilities

Average (Arithmetic Mean)

• $\text{Average} = \frac{\text{sum of all values}}{\text{number of values}}$
• A useful rearrangement: sum = average × number of values. If the ACT tells you the average of 5 tests is 88, the total sum is $88 \times 5 = 440$. This is often the real key to solving average problems.

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Quick Reference Table

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Self-Check Questions

1. What do the distance formula and the Pythagorean theorem have in common, and why does this connection make sense geometrically?

2. If you double the radius of a circle, what happens to its circumference? What happens to its area? Why are these results different?

3. Which two volume formulas follow the pattern "base area × height," and how does recognizing this pattern help you with unfamiliar 3D shapes?

4. You see a triangle with angles of 30°, 60°, and 90°. Without calculating, what is the ratio of the shortest side to the longest side?

5. When would you use the slope formula versus the distance formula if given two coordinate points? What question is each formula answering?

6. You know the average score on 4 tests is 85. How do you find the total number of points scored, and why is that rearrangement useful?