## 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 your high school 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 learn, memorize and understand 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.

Here’s a list of the most important formulas in each section.

**Pre-Algebra / Elementary Algebra **

These algebra 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

## The 31 Critical ACT Math Formulas You MUST Know

### Algebra

#### Linear Equations & Functions

There will be at least five to six questions on linear equations and functions on every ACT test, so this is a very important section to know.

**Slope**

**Slope-Intercept Form**

- A linear equation is written as y=mx+b
*m*is the slope and*b*is the y-intercept (the point of the line that crosses the y-axis)- A line that passes through the origin (y-axis at 0), is written as y=mx
- If you get an equation that is NOT written this way (i.e. mx−y=b), re-write it into y=mx+b

**Midpoint Formula**

- Given two points, A(x
_{1},y_{1}), B(x_{2},y_{2}), find the midpoint of the line that connects them:

**You don’t actually need this formula,** as you can simply graph your points and then create a right triangle from them. The distance will be the hypotenuse, which you can find via the pythagorean theorem

### Logarithms

**There will usually only be one question on the test involving logarithms.** If you’re worried about having to memorize too many formulas, don’t worry about logs unless you’re trying for a perfect score.

## Statistics and Probability

### Averages

The average is the same thing as the mean

- Find the average/mean of a set of terms (numbers)

### Probabilities

Probability is a representation of the odds of something happening. A probability of 1 is guaranteed to happen. A probability of 0 will never happen.

### Combinations

The possible amount of different combinations of a number of different elements

- A “combination” means the order of the elements doesn’t matter (i.e. a fish entree and a diet soda is the same thing as a diet soda and a fish entree)
- Possible combinations = number of element A * number of element B * number of element C….
- e.g. In a cafeteria, there are 3 different dessert options, 2 different entree options, and 4 drink options. How many different lunch combinations are possible, using one drink, one, dessert, and one entree?
- The total combinations possible = 3 * 2 * 4 = 24

### Percentages

- Find
*x*percent of a given number*n*

## Geometry

**Rectangles**

**Rectangular Solid**

**Volume**

Volume=lwh

*h*is the height of the figure

### Parallelogram

An easy way to get the area of a parallelogram is to drop down two right angles for heights and transform it into a rectangle.

- Then solve for
*h*using the pythagorean theorem

**Area**

**Area=lh**

- (This is the same as a rectangle’s
*lw*. In this case the height is the equivalent of the width)

### Triangles

*b*is the length of the base of triangle (the edge of one side)*h*is the height of the triangle- The height is the same as a side of the 90 degree angle in a right triangle. For non-right triangles, the height will drop down through the interior of the triangle, as shown in the diagram.

**Pythagorean Theorem**

a^{2}+b^{2}=c^{2}

- In a right triangle, the two smaller sides (a and b) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle)

**Properties of Special Right Triangle: Isosceles Triangle**

- An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.
- An isosceles right triangle always has a 90 degree angle and two 45 degree angles.
- The side lengths are determined by the formula:
*x, x, x*√2, with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides * √2.- E.g., An isosceles right triangle may have side lengths of 12, 12, and 12√2.

**Properties of Special Right Triangle: 30, 60, 90 Degree Triangle**

- A 30, 60, 90 triangle describes the degree measures of its three angles.
- The side lengths are determined by the formula:
*x*,*x*√3, and 2*x*.- The side opposite 30 degrees is the smallest, with a measurement of
*x.* - The side opposite 60 degrees is the middle length, with a measurement of
*x*√3. - The side opposite 90 degree is the hypotenuse, with a length of 2
*x.* - For example, a 30-60-90 triangle may have side lengths of 5, 5√3, and 10.

- The side opposite 30 degrees is the smallest, with a measurement of

### Trapezoids

**Area**

- Take the average of the length of the parallel sides and multiply that by the height.

- Often, you are given enough information to drop down two 90 angles to make a rectangle and two right triangles. You’ll need this for the height anyway, so you can simply find the areas of each triangle and add it to the area of the rectangle, if you would rather not memorize the trapezoid formula.
- Trapezoids and the need for a trapezoid formula
**will be at most one question on the test**. Keep this as a minimum priority if you’re feeling overwhelmed.

### Circles

**Area**

Area=πr^{2}

*π*is a constant that can, for the purposes of the ACT, be written as 3.14 (or 3.14159)- Especially useful to know if you don’t have a calculator that has a π feature or if you’re not using a calculator on the test.

*r*is the radius of the circle (any line drawn from the center point straight to the edge of the circle).

**Area of a Sector**

- Given a radius and a degree measure of an arc from the center, find the area of that sector of the circle.
- Use the formula for the area multiplied by the angle of the arc divided by the total angle measure of the circle.

**Circumference**

Circumference=2πr

or

Circumference=πd

*d*is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.

**Length of an Arc**

- Given a radius and a degree measure of an arc from the center, find the length of the arc.
- Use the formula for the circumference multiplied by the angle of the arc divided by the total angle measure of the circle (360).

**An alternative to memorizing the “formulas” for arcs** is to just stop and think about arc circumferences and arc areas logically.

- If you know the formulas for the area/circumference of a circle and you know how many degrees are in a circle, put the two together.

The concept is exactly the same as the formula, but it may help you to think of it this way instead of as a “formula” to memorize.

**Equation of a Circle**

- Useful to get a quick point on the ACT, but don’t worry about memorizing it if you feel overwhelmed;
**it will only ever be worth one point.** - Given a radius and a center point of a circle (h,k)

Almost all the trigonometry on the ACT can be boiled down to a few basic concepts

### SOH, CAH, TOA

Sine, cosine, and tangent are graph functions

- The sine, cosine, or tangent of an angle (theta, written as Θ) is found using the sides of a triangle according to the mnemonic device SOH, CAH, TOA.

**Sine – SOH**

- Opposite = the side of the triangle directly opposite the angle Θ
- Hypotenuse = the longest side of the triangle

Sometimes the ACT will make you manipulate this equation by giving you the sine and the hypotenuse, but not the measure of the opposite side. Manipulate it as you would any algebraic equation:

**Cosine – CAH**

- Adjacent = the side of the triangle nearest the angle Θ (that creates the angle) that is not the hypotenuse
- Hypotenuse = the longest side of the triangle

**Tangent – TOA**

- Opposite = the side of the triangle directly opposite the angle Θ
- Adjacent = the side of the triangle nearest the angle Θ (that creates the angle) that is not the hypotenuse

**Cosecant, Secant, Cotangent**

- Cosecant is the reciprocal of sine

**Useful Formulas to Know**

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