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Perimeter of a Rectangle
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Introduction

In this lesson, you will review how to find the perimeter of a rectangle and a square.

Real World Application

This is an image of three angles. There is a "right angle" on the left side, which is a 90 degree angle. Under the angle a text displays which reads: "These two lines are perpendicular." On the right side, there are two angles, and a text which says "NOT right angles." These angles are not perpendicular, which means they are not 90 degree angles.

A right angle is a 90 degree angle. You need to know this for this lesson.


This video illustrates the lesson material below. Watching the video is optional.


What is Perimeter?

Here are some definitions to help you with this lesson:

  • Perimeter: The sum of the lengths of all sides of a shape. 
  • Rectangle: A four-sided shape whose opposite sides are the same length and all the corners form right angles.
This image shows that a rectangle has opposite sides that are the same length and all the corners form right angles.

Figure 1

  • Square: A four-sided rectangle where all sides are the same length and all the corners form right angles.

Perimeter is the sum of the lengths of all sides of a shape. For rectangles and squares, you can simplify this because some of the side have the same length.

Example 1: Perimeter of a Rectangle

\begin{align*}P &= l + l + w + w \\P &=2l + 2w &\color{red}\small\text{Combining like terms}\\\end{align*}

Example 2: Perimeter of a Square

\begin{align*}P &= w + w + w + w \\P &= 4w &\color{red}\small\text{Combining like terms}\\\end{align*}

Note: P = Perimeter, l = length, and w = width

Example 3: Solving for Perimeter

A rectangle is 100 centimeters by 140 centimeters. What is the perimeter of the rectangle?

This image shows a rectangle that has sides that are 100 centimeters by 140 centimeters.

Figure 2

Figure 2 shows what a rectangle with a width of 100 centimeters and length of 140 centimeters looks like. The formulas for finding the perimeter of this rectangle are \(P = l + l + w +w \) or \(P = 2l + 2w\).

Below shows what the perimeter formulas look like using the two different ways:

\begin{align*}P &= l + l + w + w &\color{red}\small\text{First Formula} \\P &= 140 + 140 + 100 + 100 &\color{red}\small\text{Substitute given sides}\\P &= 480cm &\color{red}\small\text{Add}\\\end{align*}

Or

\begin{align*}P &= 2l + 2w &\color{red}\small\text{Second Formula} \\P &= 2(140) + 2(100) &\color{red}\small\text{Substitute given sides}\\P &= 280 + 200&\color{red}\small\text{Multiply}\\P &= 480cm &\color{red}\small\text{Add}\\\end{align*}

Example 4: Solving for Perimeter
A square is a special kind of rectangle where all of the sides have the same length.

A field is 300 feet by 300 feet. How much fence is needed to surround this field?

A square that is 300 feet by 300 ft.

Figure 3

First, decide length and width: all sides are 300 feet. Because the length is equal to the width, you only have to use one or the other in the equation:

\begin{align*} Perimeter = w+w+w+w \end{align*}

Since the figure is a square, you know that \(l=w\), so the perimeter for any square is:

\begin{align*} Perimeter =4w\end{align*}

You can use one of two methods:

\begin{align*}P &= w + w + w + w &\color{red}\small\text{First Formula}\\P &=300 + 300 + 300 + 300 &\color{red}\small\text{Substitute given sides}\\P &= 1200ft &\color{red}\small\text{Add}\\\end{align*}

Or

\begin{align*}P &= 4w &\color{red}\small\text{Second Formula}\\P &= 4(300) &\color{red}\small\text{Substitute given sides}\\P &= 1200ft &\color{red}\small\text{ Multiply}\\\end{align*}

Both formulas will give the same answer for the perimeter of a square, which is 1200 feet in this example.


Things to Remember


  • To calculate the perimeter of a rectangle or a square, add the lengths and the widths together. 
  • The formula for a square can be simplified because all the sides are the same.

Practice Problems

  1. The top of a pizza box forms a square where each side is 35 cm. Find the perimeter of the square defined by this pizza box. (
    Video Solution
    )
  2. A rectangular swimming pool has a length of 20 ft and a width of 12 ft. Find the perimeter of the rectangle defined by this swimming pool. (
    Solution
    x
    Solution: 64 ft
    Details:
    Here is a rectangle that represents the swimming pool in this question.
    A picture of a rectangle that is 20 feet wide and 12 ft high.

    The opposite sides of a rectangle are the same lengths. This means there are actually two sides that are \({\color{Blue}20}\) feet long and two sides that are \({\color{Green}12}\) feet long.
    This is the same rectangle as previously shown, but this time all the sides are labeled with the corresponding width and height.

    Since all the sides are measured using the same units, all you need to do to find the perimeter of the pool is to add together the lengths of all four sides.

    \({\color{Green}12} + {\color{Green}12} + {\color{Blue}20} + {\color{Blue}20} = 64\)

    Another way to add up the sides of the pool is to multiply the lengths of the sides by 2 and add them together since there are two sides of each length.

    \(2 ({\color{Green}12}) + 2 ({\color{Blue}20}) = {\color{Green}24} + {\color{Blue}40} = 64\)

    The total perimeter of the pool is 64 feet.
    )
  3. The borders for the state of Colorado form a rectangle that measures about 360 miles long and 280 miles wide. Find the perimeter of the rectangle defined by this state. (
    Video Solution
    x
    )
  4. A rectangular flag has a length of 150 cm and a width of 120 cm. Find the perimeter of the rectangle defined by this flag. (
    Solution
    x
    Solution: 540 cm
    )
  5. The cover of a book has a length of 5 in and a width of 8 in. Find the perimeter of the rectangle defined by this book cover. (
    Solution
    x
    Solution: 26 in
    Details:
    The sides of this book have all been measured in inches.
    A rectangle with the words 8 inches written to the right and 5 inches written on top representing the side lengths of the book. The words, “A Book,” are written inside the rectangle.

    The opposite sides of a rectangle have the same lengths. This means this rectangle has two sides that measure \({\color{Green}5}\)inches and two sides that measure \({\color{Blue}8}\) inches.
    This is the same rectangle as previously used in this solution except now all the sides are labeled. Below the rectangle, there is now written 5 inches, and to the left is written 8 inches.

    The total perimeter is found by adding up the lengths of all the sides.

    \({\color{Green}5} + {\color{Blue}8} + {\color{Green}5} + {\color{Blue}8} = {\color{Green}5} + {\color{Green}5} + {\color{Blue}8} + {\color{Blue}8} = 26\)

    (The order in which you add up the sides doesn’t matter.)

    Another way to think of this is to multiply each side length by 2 since there are two sides of the same length for each length, and then add these together.

    \(2 ({\color{Green}5}) + 2 ( {\color{Blue}8} ) = {\color{Green}10} + {\color{Blue}16} = 26\)

    The perimeter of the rectangle defined by this book is 26 inches.
    )
  6. A basketball court is rectangular in shape with a length of 29 m and a width of 15 m. Find the perimeter of the rectangle defined by this court. (
    Solution
    x
    Solution: 88 m
    )

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