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Formulas to Know for ACT Math: The Complete Student Cheat Sheet

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🧮 Must-Know ACT Math Formulas: Your No-Fluff Survival Guide

If you’re prepping for the ACT, you’ve probably realized this: they don’t give you a formula sheet.
Unlike the SAT, where certain formulas are handed to you at the beginning of the math section, the ACT expects you to know everything cold.

That’s why learning and memorizing the most important ACT math formulas is not just helpful—it’s essential if you want a top score.

In this guide, we’ll break down:

  • The must-know formulas by category

  • Tips to remember and apply them fast

  • What formulas not to stress about

  • A downloadable list you can start reviewing now

Let’s go. 🎯

🎓 Do You Really Need to Memorize Formulas for the ACT?

Yes. Here’s why:

  • You don’t get a reference sheet.

  • Time is tight: knowing a formula instantly = faster solving.

  • Many ACT Math questions hinge on simple formula recall.

The good news? You don’t need to memorize hundreds. Just the 30–40 core formulas that show up again and again.

Essential Formulas to Know for the ACT

Master these key formulas to boost your ACT Math score

🔍
Algebra
Geometry
Trigonometry
Statistics
All Formulas

Quadratic Formula

Algebra
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The quadratic formula is used to solve quadratic equations in the form ax² + bx + c = 0, where a ≠ 0.
a, b, c: coefficients in the quadratic equation ax² + bx + c = 0
Example Question:

Solve the quadratic equation: 2x² - 5x - 3 = 0

Using the quadratic formula with a = 2, b = -5, and c = -3:

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}

x = \frac{5 \pm \sqrt{25 + 24}}{4}

x = \frac{5 \pm \sqrt{49}}{4}

x = \frac{5 \pm 7}{4}

x = \frac{5 + 7}{4} = 3 or x = \frac{5 - 7}{4} = -\frac{1}{2}

The solutions are x = 3 and x = -\frac{1}{2}

Slope Formula

Algebra
m = \frac{y_2 - y_1}{x_2 - x_1}
The slope formula calculates the steepness and direction of a line by finding the ratio of the vertical change to the horizontal change between two points on the line.
m: slope
(x₁, y₁): coordinates of the first point
(x₂, y₂): coordinates of the second point
Example Question:

Find the slope of the line passing through the points (3, 7) and (5, 11).

Using the slope formula with (x₁, y₁) = (3, 7) and (x₂, y₂) = (5, 11):

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 7}{5 - 3} = \frac{4}{2} = 2

The slope of the line is 2.

Distance Formula

Algebra
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
The distance formula calculates the distance between two points in a coordinate plane using the Pythagorean theorem.
d: distance between the points
(x₁, y₁): coordinates of the first point
(x₂, y₂): coordinates of the second point
Example Question:

Find the distance between the points (-2, 3) and (4, 7).

Using the distance formula with (x₁, y₁) = (-2, 3) and (x₂, y₂) = (4, 7):

d = \sqrt{(4 - (-2))^2 + (7 - 3)^2}

d = \sqrt{6^2 + 4^2}

d = \sqrt{36 + 16}

d = \sqrt{52}

d = 2\sqrt{13} ≈ 7.21

The distance between the points is 2\sqrt{13} units.

Midpoint Formula

Algebra
\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
The midpoint formula finds the coordinates of the point exactly halfway between two given points.
(x₁, y₁): coordinates of the first point
(x₂, y₂): coordinates of the second point
Example Question:

Find the midpoint of the line segment with endpoints at (6, -2) and (-4, 8).

Using the midpoint formula with (x₁, y₁) = (6, -2) and (x₂, y₂) = (-4, 8):

Midpoint = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Midpoint = \left(\frac{6 + (-4)}{2}, \frac{-2 + 8}{2}\right)

Midpoint = \left(\frac{2}{2}, \frac{6}{2}\right)

Midpoint = (1, 3)

The midpoint of the line segment is at (1, 3).

Compound Interest

Algebra
A = P\left(1 + \frac{r}{n}\right)^{nt}
The compound interest formula calculates the final amount when interest is compounded a certain number of times per year.
A: final amount
P: principal (initial investment)
r: annual interest rate (in decimal form)
n: number of times interest is compounded per year
t: time in years
Example Question:

If $2,000 is invested at 6% interest compounded quarterly, what will be the balance after 3 years?

Using the compound interest formula with P = $2,000, r = 0.06, n = 4, and t = 3:

A = P\left(1 + \frac{r}{n}\right)^{nt}

A = 2000\left(1 + \frac{0.06}{4}\right)^{4 \times 3}

A = 2000\left(1 + 0.015\right)^{12}

A = 2000 \times (1.015)^{12}

A = 2000 \times 1.196

A = $2,392

The balance after 3 years will be $2,392.

Area of a Triangle

Geometry
A = \frac{1}{2} \times b \times h
This formula calculates the area of a triangle using its base and height.
A: area of the triangle
b: length of the base
h: height (perpendicular distance from the base to the opposite vertex)
Example Question:

A triangle has a base of 8 inches and a height of 5 inches. What is its area?

Using the area formula with b = 8 and h = 5:

A = \frac{1}{2} \times b \times h

A = \frac{1}{2} \times 8 \times 5

A = \frac{1}{2} \times 40

A = 20

The area of the triangle is 20 square inches.

Area of a Circle

Geometry
A = \pi r^2
This formula calculates the area of a circle using its radius.
A: area of the circle
r: radius of the circle
π: pi (approximately 3.14159)
Example Question:

What is the area of a circle with a radius of 6 cm?

Using the area formula with r = 6:

A = \pi r^2

A = \pi \times 6^2

A = \pi \times 36

A = 36\pi

A ≈ 113.1 cm²

The area of the circle is 36π or approximately 113.1 square centimeters.

Pythagorean Theorem

Geometry
a^2 + b^2 = c^2
The Pythagorean theorem relates the sides of a right triangle. The sum of the squares of the two legs equals the square of the hypotenuse.
a, b: lengths of the legs (the two sides that form the right angle)
c: length of the hypotenuse (the side opposite the right angle)
Example Question:

If a right triangle has legs of lengths 5 and 12, what is the length of the hypotenuse?

Using the Pythagorean theorem with a = 5 and b = 12:

a^2 + b^2 = c^2

5^2 + 12^2 = c^2

25 + 144 = c^2

169 = c^2

c = \sqrt{169} = 13

The length of the hypotenuse is 13 units.

Volume of a Cylinder

Geometry
V = \pi r^2 h
The volume of a cylinder is calculated using the area of the circular base multiplied by the height.
V: volume of the cylinder
r: radius of the circular base
h: height of the cylinder
π: pi (approximately 3.14159)
Example Question:

Find the volume of a cylinder with a radius of 4 inches and a height of 10 inches.

Using the volume formula with r = 4 and h = 10:

V = \pi r^2 h

V = \pi \times 4^2 \times 10

V = \pi \times 16 \times 10

V = 160\pi

V ≈ 502.65 in³

The volume of the cylinder is 160π or approximately 502.65 cubic inches.

Surface Area of a Rectangular Prism

Geometry
SA = 2(lw + lh + wh)
The surface area of a rectangular prism is the sum of the areas of all six faces.
SA: surface area
l: length
w: width
h: height
Example Question:

Find the surface area of a rectangular prism with length 5 cm, width 3 cm, and height 2 cm.

Using the surface area formula with l = 5, w = 3, and h = 2:

SA = 2(lw + lh + wh)

SA = 2(5 × 3 + 5 × 2 + 3 × 2)

SA = 2(15 + 10 + 6)

SA = 2(31)

SA = 62

The surface area of the rectangular prism is 62 square centimeters.

Trigonometric Ratios (SOH-CAH-TOA)

Trigonometry
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \\ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \\ \tan \theta = \frac{\text{opposite}}{\text{adjacent}}
These three ratios define the basic trigonometric functions for right triangles. The mnemonic SOH-CAH-TOA helps remember them.
θ (theta): the angle of interest
opposite: length of the side opposite to the angle
adjacent: length of the side adjacent to the angle
hypotenuse: length of the longest side (opposite the right angle)
Example Question:

In a right triangle, the hypotenuse is 10 cm and one angle is 30°. What is the length of the side opposite to the 30° angle?

We can use the sine ratio since we need to find the opposite side given the hypotenuse:

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}

\sin 30° = \frac{\text{opposite}}{10}

We know that \sin 30° = \frac{1}{2}

\frac{1}{2} = \frac{\text{opposite}}{10}

opposite = 10 × \frac{1}{2} = 5

The length of the side opposite to the 30° angle is 5 cm.

Law of Sines

Trigonometry
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
The Law of Sines relates the sides of a triangle to the sines of the opposite angles. It's useful for solving triangles when you know one side and two angles, or two sides and the angle opposite one of them.
A, B, C: angles of the triangle
a, b, c: lengths of the sides opposite to angles A, B, and C, respectively
Example Question:

In triangle ABC, if angle A = 45°, angle B = 60°, and side a = 12 cm, find the length of side b.

First, find angle C using the fact that angles in a triangle sum to 180°:

A + B + C = 180°

45° + 60° + C = 180°

105° + C = 180°

C = 75°

Now, use the Law of Sines to find side b:

\frac{\sin A}{a} = \frac{\sin B}{b}

\frac{\sin 45°}{12} = \frac{\sin 60°}{b}

b = \frac{12 \times \sin 60°}{\sin 45°}

b = \frac{12 \times \frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}}

b = 12 \times \frac{\sqrt{3}}{2} \times \sqrt{2}

b = 12 \times \frac{\sqrt{6}}{2}

b = 6\sqrt{6} ≈ 14.7 cm

The length of side b is 6\sqrt{6} or approximately 14.7 cm.

Law of Cosines

Trigonometry
c^2 = a^2 + b^2 - 2ab\cos C \\ b^2 = a^2 + c^2 - 2ac\cos B \\ a^2 = b^2 + c^2 - 2bc\cos A
The Law of Cosines generalizes the Pythagorean theorem to any triangle. It's useful for solving triangles when you know two sides and the included angle, or when you know all three sides.
a, b, c: lengths of the sides of the triangle
A, B, C: angles opposite to sides a, b, and c, respectively
Example Question:

In triangle ABC, if a = 8, b = 12, and angle C = 60°, find the length of side c.

Using the Law of Cosines:

c^2 = a^2 + b^2 - 2ab\cos C

c^2 = 8^2 + 12^2 - 2(8)(12)\cos 60°

c^2 = 64 + 144 - 2(96)(\frac{1}{2})

c^2 = 64 + 144 - 96

c^2 = 112

c = \sqrt{112} = 4\sqrt{7} ≈ 10.58

The length of side c is 4\sqrt{7} or approximately 10.58 units.

Special Angle Values

Trigonometry
\sin 0° = 0, \quad \sin 30° = \frac{1}{2}, \quad \sin 45° = \frac{1}{\sqrt{2}}, \quad \sin 60° = \frac{\sqrt{3}}{2}, \quad \sin 90° = 1 \\ \cos 0° = 1, \quad \cos 30° = \frac{\sqrt{3}}{2}, \quad \cos 45° = \frac{1}{\sqrt{2}}, \quad \cos 60° = \frac{1}{2}, \quad \cos 90° = 0
Memorizing these special angle values for sine and cosine can save time during the ACT, as they appear frequently in problems.
Tangent values can be derived using the formula: tan θ = sin θ / cos θ
Example Question:

If sin θ = 1/2, what is the value of cos θ?

We know that sin θ = 1/2, which means θ = 30° (or θ = 150°, but the ACT typically works with angles in the first quadrant unless otherwise specified).

For θ = 30°, we know that cos 30° = \frac{\sqrt{3}}{2}

Therefore, cos θ = \frac{\sqrt{3}}{2} ≈ 0.866

Mean (Average)

Statistics
\text{Mean} = \frac{\sum x_i}{n}
The mean (or average) is calculated by adding up all the values in a data set and dividing by the number of values.
x₁, x₂, ..., xₙ: individual values in the data set
n: number of values in the data set
Example Question:

Find the mean of the following test scores: 78, 84, 92, 65, 89, 72.

Using the mean formula:

Mean = \frac{78 + 84 + 92 + 65 + 89 + 72}{6}

Mean = \frac{480}{6}

Mean = 80

The mean (average) of the test scores is 80.

Median

Statistics
\text{Median} = \begin{cases} x_{\frac{n+1}{2}} & \text{if } n \text{ is odd} \\ \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} & \text{if } n \text{ is even} \end{cases}
The median is the middle value in a data set when the values are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values.
Example Question:

Find the median of the following data set: 12, 5, 22, 19, 8, 15, 27.

First, arrange the data in ascending order: 5, 8, 12, 15, 19, 22, 27

Since there are 7 values (an odd number), the median is the middle value (the 4th value).

Median = 15

Probability

Statistics
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
Probability measures the likelihood of an event occurring. It is expressed as a number between 0 (impossible) and 1 (certain).
Example Question:

A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If a marble is drawn at random, what is the probability of drawing a blue marble?

Number of favorable outcomes (blue marbles) = 3

Total number of possible outcomes (all marbles) = 4 + 3 + 5 = 12

P(blue) = \frac{3}{12} = \frac{1}{4} = 0.25

The probability of drawing a blue marble is \frac{1}{4} or 0.25 or 25%.

Standard Deviation

Statistics
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
s: standard deviation
x₁, x₂, ..., xₙ: individual values in the data set
x̄: mean of the data set
n: number of values in the data set
Note: On the ACT, you typically won't need to calculate standard deviation from scratch, but you should understand what it represents.
Example Question:

Which data set has a larger standard deviation?
Data Set A: 10, 10, 10, 10, 10
Data Set B: 5, 8, 10, 12, 15

For Data Set A, all values are the same (10), so there is no variation from the mean. The standard deviation is 0.

For Data Set B, the values are spread out from the mean (which is 10), so there is variation. The standard deviation is greater than 0.

Therefore, Data Set B has a larger standard deviation.

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