What Formulas Do You Need to Know for the ACT Math Test Part 2

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Areas and Volumes

On the ACT Math section, you will need to be able to calculate the area and volume of certain shapes. Use these equations:

Area of a trapezoid:

Area of a Trapezoid

Area of a parallelogram:

Area of a Parallelogram

Volume of a prism:

Volume of a Prism

Volume of a cone:

Volume of a Cone

Volume of a sphere:

Volume of a Sphere

Right Triangles

Area of a Right Triangle

You can calculate the area of a right triangle using this equation:

Area of a right triangle:

Area of a Right Triangle

Want to see some examples in action? View how to calculate the area of a triangle.

Pythagorean Theorem

The Pythagorean Theorem can be used to solve for the length of a side on a right triangle. A right triangle is formed when the two short sides (a and b) are perpendicular to one another. The third side, c, is called the hypotenuse.

The Pythagorean Theorem:

This Pythagorean Theorem lesson goes into more detail on how the theorem can be used to find distances. Use Pythagorean Theorem practice to apply it!

SOHCAHTOA

Trigonometry can help calculate information about the lengths of sides and the size of angles in a right triangle. There are three functions you can use for this: sine, cosine and tangent.

Where:

  • is the angle in the question
  • O is the side opposite that angle
  • H is the hypotenuse
  • A is the adjacent side (the side next to the angle)

This can be hard to remember, so use the mnemonic device “SOHCAHTOA” to help.

  • SOH = Sine is Opposite over Hypotenuse
  • CAH = Cosine is Adjacent over Hypotenuse
  • TOA = Tangent is Opposite over Adjacent

Non-Right Triangles

Sine Law

If the given triangle doesn՚t have a right angle, use the “Law of Sines” (or the Sine Law) to work out the size of angles or the lengths of the sides by using ratios. If you are given two angles and one side, the Sine Law can find the missing side.

Label the angles as A, B and C. Then label the sides as follows: side a is opposite angle A, side b is opposite angle B and side c is opposite angle C.

Cosine Law

The “Law of Cosines” , or the Cosine Law, can be used to work out the length of a missing side or angle of any kind of triangle.

To use the Cosine Law to find the length of a missing side, you need to know:

  • The lengths of the other two sides
  • The size of the angle opposite the side you want to find

To work out the size of an angle, you need to know the measurements of all three sides of the triangle.

Area of Non-Right Triangles

Work out the area of any triangle, not just right triangles, using the following formula:

Area of Non-Right Triangles

Where A is the area, a and b are the sides either side of the angle C. This can also be used to work out whether two triangles are congruent (identical) or similar (in proportion) to each other.

Trigonometric Identities

The four basic trig identities are also known as Pythagorean identities. They are as follows:

These can be useful when simplify expressions with trig functions in them.

Formulas for an X-Y Axis

You will need to know a few different formulas for the x-y axis, also known as the Cartesian coordinate plane. These formulas can be used to solve problems with graphs.

Slope of a Graph

The slope, also known as the gradient, of a graph tells you how steep a line is. The variable m in this equation stands for “slope,” which is defined as ‘change in y’ over ‘change in x’ . Written as an equation this looks like:

Slope of a Graph

If you are unfamiliar with , or ‘delta,’ it means ‘change in’ . Sometimes it helps to remember this as ‘rise over run’ — how much the line rises over how much it runs along the bottom.

Equation of a Line

All straight-line graphs have the equation:

Where m is the slope of the graph and b is where the line crosses the y axis, also known as the ‘y intercept’ . Use this information to plot a function onto a set of axes or, if given a straight line graph, use it to write out the graph՚s function.

Learn more about how to understand and apply straight line graphs in Creating and Interpreting Straight-Line & Line Segment Graphs.

The Midpoint Formula

The formula for the midpoint of a line segment can be used to work out where the middle of any line segment on any Cartesian coordinate plane (any x-y plane) . It gives an answer as a coordinate.

Midpoint

The Midpoint Formula

Where d is the distance and the co-ordinates are and

Graph of a Circle

There is a special equation used for graphing circles. The formula is:

Use the following characteristics to figure out if an equation is a circle. If it is a circle:

  • x and y terms are squared
  • All terms in the equation are positive
  • The center point of the circle is (h, k)
  • r represents the radius of the circle

This information should help you sketch a graph. Write the formula for the circle when given the plot or work out the midpoint of the circle from the formula. Sometimes h and k will not be in this order, but h will always be the number in the with x and k will always be the number with y.

Graph of an Ellipse

An ellipse is like a circle that has been stretched. You can describe an ellipse using this equation:

Where:

  • (h, k) is the center of the ellipse,
  • a is the radius in the horizontal, or x, direction,
  • b is the radius in the vertical, or y, direction.

There is always an addition sign in between the x and y parts of the equation, and the equation always equals 1.

Fill in the variables to sketch an ellipse from the equation or use a plot of an ellipse to write the equation.

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