# Perimeter of Squares and Rectangles

## Find total edge measures of rectilinear quadrilaterals.

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Perimeter of Squares and Rectangles

Michael's family just got a new puppy. The first thing they want to do is put a fence around their backyard so they can keep their puppy contained. It is Michael's job to figure out how much fencing they will need to his dad can go buy the materials. Michael needs to figure out how he can measure the backyard to find the amount of fencing they will need.

Their backyard is rectangular shaped with dimensions of 50 feet wide and 70 feet long.

In this concept, you will learn how to calculate the space around an area, which is called the perimeter.

### Perimeter

The perimeter is the distance around the edge of an object. We can find the perimeter of any figure. In this concept, you will learn how to find the perimeter of squares and rectangles.

The diagram above shows a square with a side length of 5 feet. Notice that it only gives one side dimension, but remember that all sides are equal in a square, so only one side is labeled to account for all sides.

The side length of a square is used to calculate the perimeter of the square. The formula for calculating the perimeter of a square is to add up all the side lengths. In the equation, P\begin{align*}P\end{align*} represents perimeter and s\begin{align*}s\end{align*} represents side length.

P=s+s+s+s\begin{align*}P=s+s+s+s\end{align*}

This square has a side length of 5 feet, so you would add the 4 equal sides measured at 5 feet together.

PP=5+5+5+5=20ft\begin{align*}P&=5+5+5+5\\ P&=20 ft\end{align*}

There is another shortcut way for calculating the perimeter of a square as well.

P=4s\begin{align*}& P=4\cdot s\\ \end{align*}

Remember that multiplication is a shortcut for repeated addition. Since all the sides are equal in a square, you can multiply the side length by the number of sides (which is always four in a square).

PPP=4s=4(5)=20 ft\begin{align*}P &= 4s\\ P &= 4(5)\\ P &= 20 \ ft\end{align*}

The formula for finding the perimeter of a rectangle is very similar, however the shortcut formula is different since a rectangle does not always have 4 equal sides.

This is the formula for finding the perimeter of a rectangle.

P=2l+2w\begin{align*}P=2l+2w\end{align*}

Since we have two lengths that have the same measure and two widths that have the same measure, we can add two times the length plus two times the width, and that will give us the distance around the rectangle.

This rectangle has a length of 8 inches and a width of 6 inches (notice the label next to the measurements are now inches, not feet), we can substitute these dimensions into our formula and solve for the perimeter of the rectangle.

PPPP=2l+2w=2(8)+2(6)=16+12=28 inches\begin{align*}P &= 2l+2w\\ P &= 2(8)+2(6)\\ P &= 16+12\\ P &= 28 \ inches\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Michael and his family's new puppy.

They want to put a fence around their backyard to contain the puppy. The dimensions of their rectangular backyard are 50 feet wide by 70 feet long. Michael can use the formula for perimeter of a rectangle to find the total amount of fencing his family will need in their backyard.

First, Michael writes out the formula for perimeter of a rectangle.

P=2l+2w\begin{align*}P= 2l+2w\end{align*}

Next, Michael plugs in the dimensions of his backyard.

P=2(70)+2(50)\begin{align*}P= 2(70) + 2(50)\end{align*}

Then, Michael solves the equation by multiplying the length and multiplying the width then adding them together.

PP=140+100=240 ft\begin{align*}P&= 140+100\\ P&= 240\ ft\end{align*}

The answer is Michael's family will need 240 ft. of fencing for their backyard.

#### Example 2

Find the perimeter of a rectangle with dimensions of 12 feet by 15 feet.

First, write out the correct formula for the perimeter of a rectangle

P=2l+2w\begin{align*}P= 2l+2w\end{align*}

Next, plug your dimensions for the problem you are solving into the formula to create your equation.

P=2(12)+2(15)\begin{align*}P= 2(12)+2(15)\end{align*}

Then, solve the equation using order of operations. Multiply the length and width first, then add the two numbers together.

PP=24+30=54ft\begin{align*}P&= 24+30\\ P&=54 ft\end{align*}

The answer is 54 feet is the perimeter of this rectangle.

#### Example 3

Find the perimeter of a square with a side length of 7 inches.

First, write out the correct formula for the figure.

\begin{align*}P&= 4s\\ &or\\ P&= s+s+s+s\end{align*}

Next, plug the dimensions into the formula.

\begin{align*}P&= 4(7)\\ &or\\ P&=7+7+7+7\end{align*}

Then, solve the equation by completing the calculation.

\begin{align*}P= 28 in\end{align*}

#### Example 4

Find the perimeter of a rectangle with a length of 9 feet and a width of 3 feet.

First, write out the formula for perimeter of a rectangle.

\begin{align*}P= 2l + 2w\end{align*}

Next, plug the dimensions into the formula to create your equation.

\begin{align*}P= 2(9) + 2(3)\end{align*}

Then, solve the equation following order of operations and multiplying the length and width first before adding.

\begin{align*}P&= 18+6\\ P&= 24\ ft\end{align*}

#### Example 5

Find the perimeter of a square with a side length of 2 centimeters.

First, write the formula for perimeter of a square.

\begin{align*}P= 4s\end{align*}

Next, plug the dimensions into the formula to create your equation.

\begin{align*}P= 4(2)\end{align*}

Then, solve the equation by multiplying.

\begin{align*}P= 8cm\end{align*}

### Review

Find the perimeter of each of the following squares and rectangles.

1. A square with a side length of 6 inches.
2. A square with a side length of 4 inches.
3. A square with a side length of 8 centimeters.
4. A square with a side length of 12 centimeters.
5. A square with a side length of 9 meters.
6. A rectangle with a length of 6 inches and a width of 4 inches.
7. A rectangle with a length of 9 meters and a width of 3 meters.
8. A rectangle with a length of 4 meters and a width of 2 meters.
9. A rectangle with a length of 17 feet and a width of 12 feet.
10. A rectangle with a length of 22 feet and a width of 18 feet.
11. A square with a side length of 16 feet.
12. A square with a side length of 18 feet.
13. A square with a side length of 21 feet.
14. A rectangle with a length of 18 feet and a width of 13 feet.
15. A rectangle with a length of 60 feet and a width of 27 feet.
16. A rectangle with a length of 57 feet and a width of 22 feet.

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### Vocabulary Language: English

TermDefinition
Formula A formula is a type of equation that shows the relationship between different variables.
Perimeter Perimeter is the distance around a two-dimensional figure.
Rectangle A rectangle is a quadrilateral with four right angles.
Square A square is a polygon with four congruent sides and four right angles.