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Identify the correct formula for a given shape.

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what's covered
  1. Definition of Formulas
  2. Examples of Common Formulas
  3. Substitution in Formulas

1. Definition of Formulas

Formulas are often times thought of as special types of equations where two or more quantities are equated to one another.

term to know
A mathematical rule that relates two or more quantities.

There are several different types of formulas that we will come across when working in most math courses. Here are a few of the most common formulas.

2. Examples of Common Formulas

Area of a Rectangle
A equals b h, where A is the area, b is the base, and h is the height.


Area of a Triangle
A equals 1 half b h, where A is area, b is base, and h is height.


Area of a Circle
A equals pi space r squared


Volume of a Rectangular Prism
V equals l w h, where V is volume, l is length, w is width, and h is height


Volume of a Cylinder
V equals pi space r squared h, where V is volume, r is the radius of the circular base, h is height, and π is approximately 3.14


Volume of a Sphere
V equals 4 over 3 pi space r cubed, where V is volume, r is the radius, and π is approximately 3.14


Pythagorean Theorem
a squared plus b squared equals c squared, where a and b are the vertical and horizontal legs of the triangle, and c is the hypotenuse of the right triangle (opposite the right angle).


Compound Interest
A equals P left parenthesis 1 plus r over n right parenthesis to the power of n t end exponent, where A is the account balance, P is the principal (initial) balance, r is the interest rate (expressed as a decimal), n is the number of times per year the interest is compounded, and t is time in years.

3. Substitution in Formulas

Formulas become most handy when we are given the value of several quantities and asked to determine the value of another quantity. For example, suppose we are told that a rectangle has an area of 40 square feet and a base of 4 feet. What would the height of the rectangle be?

To calculate the length of the rectangle, we can take the area, A, and base, b, and substitute them in the formula for a rectangle’s area. We can then solve for the height, h.


When we wish to tell whether or not we can use a formula in this way, we simply need to check that we have a value of all but one variable in the formula. If that is the case then we can solve for the unknown quantity using the appropriate formula.

The definition of formulas states that formulas are used to relate two or more different quantities. There are many examples of common formulas, such as area of a rectangle or triangle, or the volume of a sphere or cylinder. We most often use formulas when we are given one or more of those variables, and we need to solve for an unknown variable, or something that we want to find. This is done by substituting values for known variables into our formula and then evaluating the formula to find the unknown variable.

Terms to Know

A mathematical statement that two expressions or quantities have the same value.


A mathematical rule that relates two or more quantities.


A symbol (usually a letter) used to represent a value that can change.

Formulas to Know
Area of Circle

A subscript c i r c l e end subscript equals pi r squared

Area of Rectangle

A subscript r e c tan g l e end subscript equals b h

Area of Triangle

A subscript t r i a n g l e end subscript equals 1 half b h

Compound Interest

A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent

Volume of Cylinder

V subscript c y l i n d e r end subscript equals pi r squared h

Volume of Rectangular Prism

V subscript r e c t. p r i s m end subscript equals l w h

Volume of Sphere

V subscript s p h e r e end subscript equals 4 over 3 pi r cubed