Introduction
In this lesson, you will review how to find the perimeter of a rectangle and a square.
Real World Application
It is important to understand the difference between a right angle, which is 90 degrees, and any other angle. As shown above, the first angle is made of perpendicular lines and is 90 degrees, which makes it a right angle. The two other angles are not made of perpendicular lines, and therefore are not right angles.
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. (In Figure 1, the first shape is a rectangle, and the other two are not.)
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?
Figure 2
Figure 2 shows a rectangle with a width of 100 centimeters and length of 140 centimeters. 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 methods:
\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?
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
- 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
- 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
- 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
- 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
- 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
- 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