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# 1.13: Find Perimeter and Area of Squares and Rectangles Using Formulas

Difficulty Level: At Grade Created by: CK-12
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Credit: vicki moore
Source: https://www.flickr.com/photos/vickimoore/3004369848/in/photolist-5zubu9-5zubf7-paUJU5-8jEM41-6fe6s3-a5D5MW-a5AbAg-a5D2mU-a5A8da-5zubxQ-rA63mp-r3utYj-r3VyBg-r9PqJX-cPQjXh-6fe5Xf-eas9yu-r1Ku7b-r1KuMu-r3XSX1-qr6juG-qFnvos-cqt6Km-fi37Ky-rivCLU-qMiHKX-qLsMVs-6UqwsG-paUELW-reBoFt-awmrCX-fg5cwK-awp9eS-qLzuvn-r3VxQX-6QUrju-qWyrws-r3TA8g-r1haVW-qqwKtz-qFSFt2-qX6dup-r3q6ir-58gybD-qLzv8V-qWDCYF-qEdkrE-qLwctL-q11AmX-qEmapR

Raj has a small patch of land that he wants to turn into a vegetable garden. However, he wants to build a small fence around it before he plants anything. He knows that the area of the land is 240 square feet and that the length of one side is 15 feet. In order to build a fence, he needs to know the width of the land too. How can Raj write and solve an equation to determine the width of his future garden?

In this concept, you will learn to find the perimeter and area of squares and rectangles using formulas.

### Guidance

A formula is a method that has been proven to work when solving specific types of problems.

Let’s explore some of those familiar formulas by looking at rectangles, squares, area and perimeter.

The perimeter of a figure is the distance around the figure. Perimeter is the sum of all of the sides in a square or rectangle. Since a rectangle has two sets of parallel sides, the formula for determining perimeter (P)\begin{align*}(P)\end{align*} of a rectangle is:

P=2L+2W

L=length\begin{align*}L=\text{length}\end{align*} and W=width\begin{align*}W=\text{width}\end{align*}

Let’s look at an example.

The rectangle above shows its dimensions. Find the perimeter.

First, substitute the values for the width (W)\begin{align*}(W)\end{align*} and the length (L)\begin{align*}(L)\end{align*} into the perimeter formula.

P=2(12)+2(9)

Next, complete the multiplication and addition to find the perimeter.

PPP===2(12)+2(9)24+1842

The perimeter of the rectangle is 42 inches.

Area is the amount of square units inside the figure. Area is found by multiplying the length×width\begin{align*}\text{length} \times \text{width}\end{align*}. The formula for finding the area of a rectangle is:

A=L×W

You can use the dimensions from the rectangle above to find the area of this rectangle.

First, fill the values for L\begin{align*}L\end{align*} and W\begin{align*}W\end{align*} into the formula for area.

AA==L×W12×9

Next, solve for the area by multiplying.

AA==12×9108

The area of the rectangle is 108 in2\begin{align*}108 \ in^2\end{align*}.

Notice that the unit of measurement for area is squared. That is because you multiplied a unit measure times itself: in×in=in2\begin{align*}in \times in = in^2\end{align*}. Area is always written in square units.

You can also find the perimeter and area of a square. Remember that a square has four equal sides given the symbol s\begin{align*}s\end{align*}. You can use the following formula for finding the perimeter of a square:

P=4s

Let’s look at an example.

A rectangle has a length of 12 feet and a perimeter of 72 feet. Write and solve an equation to determine the width of the rectangle.

First, substitute the values for the perimeter (P)\begin{align*}(P)\end{align*} and the length (L)\begin{align*}(L)\end{align*} into the perimeter formula.

72=2(12)+2W

Next, complete the multiplication.

72=24+2W

Then, subtract 24 from both sides to get your variable alone on the right side.

722448==2424+2W2W

Then, multiply both sides by the reciprocal of 2 to isolate your variable.

12×481482W===12×2W1W24

The width of the rectangle is 24 feet.

### Guided Practice

A square has a perimeter of 196 inches. Determine the length of one side of the square.

First, substitute the value for the perimeter (P)\begin{align*}(P)\end{align*} into the perimeter formula.

196=4s

Next, multiply both sides by the reciprocal of 4 to isolate your variable.

14×19611964s===14×4s1s49

### Examples

#### Example 1

Find the perimeter of the following square if the side length is 4.5 inches.

First, substitute the value for the side length into the perimeter formula.

PP==4s4(4.5)

Next, multiply by 4 to solve for the perimeter.

PP==4(4.5)18

#### Example 2

Can you find the area of the square in Example 1?

First, substitute the value for the side length into the area formula.

AA==s×s4.5×4.5

Next, multiply to solve for the area.

AA==4.5×4.520.25

The area of the square is 20.25 in2\begin{align*}20.25 \ in^2\end{align*}.

#### Example 3

A square has an area of 144 sq. meters. What is the side length?

First, substitute the value for the area into the area formula.

A144 m2==s×ss×s

Next, take the square root of the area to isolate s\begin{align*}s\end{align*}. Remember that the opposite of square is square root.

144144ss====s×ss214412

Credit: Helen Chang
Source: https://www.flickr.com/photos/drwhofaninva/7729194854/in/photolist-cM19mf-ogpXxR-6byzNe-cnsrRo-cLM5es-cfY8Jm-kxAt3g-9XUPQ7-ekKzE4-xbcg-8B8dRs-6nwvvU-rwrqNm-9ZCnEF-dpXZ3q-a7yt3M-dq5K5n-9onzYc-9ongVB-9onf74-9onxsg-qagcAW-95CJDq-e5X31n-e5X34r-6C3z87-8m3da1-9onj3F-9onxSg-9oqk11-9oqqnN-9oqjnJ-9oqyMU-9ouF48-9XLVrG-9onwsg-9onjSX-9oqoSf-9oquaW-9oqwPN-acUfz4-dWvKfN-crKsFu-9onuiX-9wXige-5uA6g2-95aPr4-d9rApE-9fvy1N-8B54sx

Remember Raj and his garden fence?

He knows that the area of the land is 240 square feet, the length of one side is 15 feet, and needs to know the width of the land.

First, substitute the values for the area and the side length into the area formula.

A240 ft2==L×W15 ft×W

Next, multiply both sides by the reciprocal of 15 in order to isolate the variable W\begin{align*}W\end{align*}.

115×2401W==115×151×W16

The width of Raj’s garden is 16 ft. Therefore the dimensions of the garden are 16 ft by 15 ft.

### Explore More

Find the area and perimeter of each square or rectangle using formulas and the given dimensions. Each problem will have two answers.

1. A square with a side length of 5 inches.

2. A rectangle with a length of 5 inches and a width of 3 inches.

3. A rectangle with a length of 8 cm and a width of 6 cm.

4. A square with a side length of 11 feet.

5. A rectangle with a length of 9 inches and a width of 4.5 inches.

6. A square with a side length of 7 feet.

7. A rectangle with a length of 12 meters and a width of 11 meters.

8. A square with a side length of 13 meters.

9. A rectangle with a length of 15 feet and a width of 8 feet.

10. A square with a side length of 12.5 feet.

Find the missing side length given the area of each square.

11. A=64 in2\begin{align*}A=64 \ in^2\end{align*}

12. A=36 in2\begin{align*}A = 36 \ in^2\end{align*}

13. A=81 m2\begin{align*}A = 81 \ m^2\end{align*}

14. A=100 in2\begin{align*}A=100 \ in^2\end{align*}

15. A=144 ft2\begin{align*}A = 144 \ ft^2\end{align*}

16. A=121 cm2\begin{align*}A=121 \ cm^2\end{align*}

17. A=4 mm2\begin{align*}A=4 \ mm^2\end{align*}

### Vocabulary Language: English

Area

Area

Area is the space within the perimeter of a two-dimensional figure.
Formula

Formula

A formula is a type of equation that shows the relationship between different variables.
Perimeter

Perimeter

Perimeter is the distance around a two-dimensional figure.
Square Unit

Square Unit

A Square Unit is a square with four sides that each measure 1 unit.

## Date Created:

Dec 19, 2012

Nov 20, 2015
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