# Mathematics

## 16 Critical Math ACT Math Practice Formulas You MUST Know

Let’s break down exactly what the Math section of the ACT consists of. There are 60 total multiple-choice questions taken from six areas of math: pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry. Thus, the scoring and what math formulas you need to know breaks down like this:

- Pre-Algebra / Elementary Algebra: 24 Questions, 24 Points
- Intermediate Algebra / Coordinate Geometry: 18 Questions, 24 Points
- Plane Geometry / Trigonometry: 18 Questions, 24 Points

Here’s the thing about the ACT math section: even with all the ACT math test prep you did, the ACT doesn’t give you a cheat sheet with all the math formulas written down on them. Therefore, it’s up to you to memorize them. But some critical ACT math formulas are required more frequently than others. These are the must-knows. While it might be tempting to just make a guess and move, it’s better if you’re prepared from the get-go.

Let’s take a look at the most important formulas per section.

## Pre-Algebra / Elementary Algebra

These formulas involve basic math and algebra. In other words, it requires the student to solve for an unknown variable.

#### 1. Arithmetic mean (average) = Sum of values / Number of values

Specifically used to calculate the mean value of a given set of numbers.

For example: (10 + 12 + 14 + 16) / 4 = 13

#### 2. Probability = Target outcomes / Total outcomes

Specifically used to calculate the chances of something occurring from a set of possible outcomes.

For example, A jar contains five blue marbles, five red marbles, and ten white marbles. What is the probability of picking a red marble at random?

5 / 20 = .25 or 25%

#### 3. Quadratic Formula: x = −b ± √b²-4ac/2a

Specifically used for determining the x-intercepts of a quadratic (parabolic) equation.

For example: A = 1, B = 4, C = 4

- x = -4 ± √4² – 4 (1)(4) / 2(1)
- x = -4 ± √ 16 – 4(4) / 2
- x = -4 ± √16 – 16 / 2
- x = -4 ± √ 0 / 2
- x = -4 / 2
- x = -2

## Intermediate Algebra / Coordinate Geometry

These formulas help calculate distances, lengths, and properties of points on a plane, as well as solve for variables in more complex algebraic expressions.

#### 4. Distance Formula: d=√(x₁ – x₂)² + (y₁ – y₂)²

Specifically calculates the distance between two points on a coordinate plane.

For example: Find the distance between points (6, 6) and (2, 3)

- d=√(6 – 2)² + (6 – 3)²
- d=√(4)² + (3)²
- d=√16 + 3
- d=√25
- d = 5

#### 5. Slope Formula: Slope = y₂ – y₁ / x₂ – x₁

Specifically calculates the slope (angle) of a line that connects two points on a plane.

For example: Coordinates = (-2, -1) (4, 3)

- s = 3 – (-1) / 4 – (-2)
- s = 4 / 6
- s = 2 / 3

#### 6. Slope Intercept: y=mx+b

Formula that defines a line on a plane, given a known slope and y-intercept.

For example: Slope = 2, Intercept point (0,3)

- y = 2x+3

#### 7. Midpoint Formula: (x₁+x₂) / 2, (y₁+y₂) / 2

Specifically calculates the midpoint between to points on a plane.

For example: Find the midpoint between (-1, 2) and (3, -6)

- (-1 + 3) / 2, (2 + -6) / 2
- 2 / 2, -4 / 2
- Midpoint (1, -2)

## Plane Geometry

Formulas for calculations attributes of geometric shapes within a plane and solving for variables based on the angles of a given shape (trigonometric identities).

#### 8. Area of Triangle: area = (1/2) (base) (height)

Specifically calculates the total area within a triangle based on the lengths of the sides.

For example: Base = 5, Height = 8

- a = 1/2 (5)(8)
- a = 1/2 (40)
- a = 20

#### 9. Pythagorean Theorem: a²+b²=c²

Used specifically to calculate the length of an unknown side of a right triangle, given two sides are known.

For example: a = 3, b = 4

- c² = 3² + 4²
- c² = 9 + 16
- c² = 25
- c = √25
- c = 5

#### 10. Area of Rectangle: area = length x width

Calculates specifically the total area within a rectangle shape.

For example: length = 5, width = 2

- a = 5 x 2
- a = 10

#### 11. Area of Parallelogram: area = base x height

Specifically calculates the total area within a parallelogram.

For example: base = 6, height = 12

- a = 6 x 12
- a = 72

### 12. Area of Circle: π * r²

Calculates specifically the total area within a circle.

For example: radius = 4

- a = π x 4²
- a = π x 16
- a = 50.24

#### 13. Circumference of Circle: circumference = 2π * r

Calculates specifically the length of the outline of a circle.

For example: radius = 7

- c = 2π x 7
- c = 43.98

## Trigonometry

Continues with the previous plane geometry section.

#### 14. Sine (SOH): Sine = opposite / hypotenuse

A trigonometric identity that represents the relative sizes of the sides of a triangle and can also be used to calculate unknown sides or angles of the triangle.

For example: opposite = 2.8, hypotenuse = 4.9

- s = 2.8 / 4.9
- s = 0.57

#### 15. Cosine (CAH): Cosine = adjacent / hypotenuse

A trigonometric identity that represents the relative sizes of the sides of a triangle and can also be used to calculate unknown sides or angles of the triangle.

For example: adjacent = 11, hypotenuse = 13

- c = 11 / 13
- c = 0.85

#### 16. Tangent (TOA): Tangent = opposite / adjacent

A trigonometric identity that represents the relative sizes of the sides of a triangle and can also be used to calculate unknown sides or angles of the triangle.

For example: opposite = 15, adjacent = 8

- t = 15 / 8
- t = 1.87