🎒ACT

ACT Math Formulas to Memorize for ACT

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TL;DR

The ACT Math section has 45 questions (41 scored) in 50 minutes. No formula sheet is provided, so you must have these formulas memorized cold. The test rewards knowing when and why to use each formula, not just plugging in numbers.

ACT Math Section: Quick Facts

45 questions total, 41 scored
50 minutes
No formula sheet provided
Topics: pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry

Two-Dimensional Area Formulas

These formulas measure the space inside flat shapes. Area always involves multiplying two dimensions, so units are always squared.

Area of a Rectangle

$A = lw$ — multiply length by width
Foundation for other formulas: triangle area is half a rectangle; 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; in obtuse triangles the height may fall outside the triangle
Common trap: the test often gives side lengths instead of height, requiring you to find height first using the Pythagorean theorem or special right triangle ratios

Area of a Circle

$A = \pi r^2$ — square the radius first, then multiply by $\pi$
If given diameter, divide by 2 before using this formula
Sector area: multiply by the fraction of the circle → $A_{\text{sector}} = \pi r^2 \times \frac{\theta}{360°}$

Compare: Rectangle vs. Triangle — both use base and height, but a triangle is exactly half. A triangle inside a rectangle sharing the same base and height always has half the rectangle's area.

Three-Dimensional Volume and Surface Area

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

Volume of a Rectangular Prism

$V = lwh$ — length × width × height
Units are cubed (cubic feet, cubic meters, etc.)

Volume of a Cylinder

$V = \pi r^2 h$ — circle area times height; same base-area × height pattern
Common applications: pipes, cans, tanks

Surface Area of a Rectangular Prism

$SA = 2(lw + lh + wh)$ — add the three different face areas, then double (opposite faces match)
Six faces total: two each of top/bottom, front/back, left/right

Surface Area of a Cylinder

$SA = 2\pi rh + 2\pi r^2$ — lateral (curved) surface plus two circular bases
The lateral area $2\pi rh$ is the circumference ( $2\pi r$ ) times height — imagine unrolling the curved surface into a flat rectangle
Open cylinder (no top): subtract one $\pi r^2$

Compare: Prism volume vs. Cylinder volume — both follow "base area × height." The only difference is the base shape. Recognizing this pattern helps you adapt to unfamiliar 3D shapes.

Circle Measurements

Everything connects back to the radius.

Circumference

$C = 2\pi r$ or $C = \pi d$
Arc length for a partial circle: $C \times \frac{\theta}{360°}$

Compare: Circumference uses $r^1$ (linear); area uses $r^2$ (two-dimensional). Double the radius → circumference doubles, area quadruples.

Coordinate Geometry Formulas

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
Positive slope: line goes up left to right; negative: goes down; zero: horizontal; undefined: vertical
Parallel lines: equal slopes
Perpendicular lines: slopes are negative reciprocals (e.g., $\frac{2}{3}$ and $-\frac{3}{2}$ )

Slope-Intercept Form

$y = mx + b$ — $m$ is slope, $b$ is the y-intercept
Reading a graph: find where the line crosses the y-axis for $b$ , then count rise/run for $m$

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
Works in reverse: given the midpoint and one endpoint, solve for the other

Distance Formula

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

Compare: Slope divides the differences (a ratio); distance squares and adds them (Pythagorean theorem). Know which operation matches which formula.

Triangle Relationships

Pythagorean Theorem

$a^2 + b^2 = c^2$ — right triangles only; $c$ is always the hypotenuse
Finding a leg: $a = \sqrt{c^2 - b^2}$
Common Pythagorean triples to memorize: 3-4-5, 5-12-13, 8-15-17 (and their multiples, e.g., 6-8-10)
Testing for a right triangle: if the equation holds for three given side lengths, the triangle is a right triangle

SOHCAHTOA (Trigonometric Ratios)

Applies to right triangles; $\theta$ is one of the two non-right angles.

SOH: $\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$
CAH: $\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: $\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$ — useful when the hypotenuse isn't involved

Special Right Triangle Ratios

45-45-90: sides in ratio $1 : 1 : \sqrt{2}$ — legs are equal; hypotenuse = leg × $\sqrt{2}$
30-60-90: sides in ratio $1 : \sqrt{3} : 2$ — shortest side opposite 30°, $\sqrt{3}$ side opposite 60°, longest side opposite 90°
Memorizing these lets you skip calculations when you recognize the triangle

Spot them in disguise: A square cut diagonally creates two 45-45-90 triangles. An equilateral triangle cut in half creates two 30-60-90 triangles.

Algebraic Problem-Solving

Quadratic Formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — solves any equation in the form $ax^2 + bx + c = 0$
The discriminant ( $b^2 - 4ac$ ): positive → two real solutions; zero → one repeated solution; negative → no real solutions
Use when factoring isn't obvious; the formula always works

Percent Change

$\text{Percent Change} = \dfrac{\text{New} - \text{Original}}{\text{Original}} \times 100$
Always divide by the original value, not the new one
Positive result = increase; negative result = decrease

Probability

$P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$
Always between 0 and 1
Two independent events both occurring: multiply their individual probabilities

Average (Arithmetic Mean)

$\text{Average} = \dfrac{\text{sum of all values}}{\text{number of values}}$
Key rearrangement: sum = average × number of values
- Example: average of 5 tests is 88 → total sum = $88 \times 5 = 440$ . This rearrangement is often the real key to solving ACT average problems.

Quick Reference Table

Concept	Key Formulas
2D Area	$A = lw$ , $A = \frac{1}{2}bh$ , $A = \pi r^2$
3D Volume	$V = lwh$ , $V = \pi r^2 h$
Surface Area	$SA = 2(lw + lh + wh)$ , $SA = 2\pi rh + 2\pi r^2$
Circle	$C = 2\pi r$ , $A = \pi r^2$
Coordinate Geometry	Slope: $\frac{y_2-y_1}{x_2-x_1}$ ; Line: $y = mx + b$ ; Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ ; Midpoint: averages
Right Triangles	$a^2 + b^2 = c^2$ ; SOHCAHTOA
Special Triangles	45-45-90: $1:1:\sqrt{2}$ ; 30-60-90: $1:\sqrt{3}:2$
Quadratics	$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
Statistics & Probability	Average = sum ÷ count; Probability = favorable ÷ total
Percent Change	$\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$

Self-Check Questions

What do the distance formula and the Pythagorean theorem have in common, and why does this connection make sense geometrically?
If you double the radius of a circle, what happens to its circumference? What happens to its area? Why are these results different?
Which two volume formulas follow the pattern "base area × height," and how does recognizing this pattern help with unfamiliar 3D shapes?
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?
When would you use the slope formula versus the distance formula given two coordinate points? What question is each formula answering?
You know the average score on 4 tests is 85. How do you find the total points scored, and why is that rearrangement useful?