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A rectangle with sides marked W and L.

The distance around this rectangle is L + W + L + W , or 2 L + 2 W . This is the perimeter , P , of the rectangle.

P = 2 L + 2 W

What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

A rectangles composed of 6 squares that is three high and two wide. The height is marked 3 and the width is marked 2.
A = 6 A = 2 · 3 A = L · W

The area is the length times the width.

The formula for the area of a rectangle is A = L W .

Properties of rectangles

Rectangles have four sides and four right ( 90 ° ) angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

P = 2 L + 2 W

The area of a rectangle is the product of the length and the width.

A = L · W

 

The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

Solution

Step 1. Read the problem.
Draw the figure and label it with the given information.
.
Step 2. Identify what you are looking for. the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it. Let P = the perimeter.
Step 4. Translate .
Write the appropriate formula. .
Substitute. .
Step 5. Solve the equation. .
.
Step 6. Check.

P 104 20 + 32 + 20 + 32 104 104 = 104
Step 7. Answer the question. The perimeter of the rectangle is 104 meters.
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The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

340 yards

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The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

220 feet

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The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?

Solution

Step 1. Read the problem.
Draw the figure and label it with the given information.
.
Step 2. Identify what you are looking for. the width of a rectangular room
Step 3. Name. Choose a variable to represent it. Let W = the width.
Step 4. Translate .
Write the appropriate formula. A = L W
Substitute. 168 = 14 W
Step 5. Solve the equation. 168 14 = 14 W 14
12 = W
Step 6. Check.

.
A = L W 168 14 12 168 = 168
Step 7. Answer the question. The width of the room is 12 feet.
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The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

26 feet

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The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

29 meters

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Find the length of a rectangle with perimeter 50 inches and width 10 inches.

Solution

Step 1. Read the problem.
Draw the figure and label it with the given information.
.
.
Step 2. Identify what you are looking for. the length of the rectangle
Step 3. Name. Choose a variable to represent it. Let L = the length.
Step 4. Translate .
Write the appropriate formula. .
Substitute. .
Step 5. Solve the equation. .

.

.

.
Step 6. Check.
.
P = 50 15 + 10 + 15 + 10 50 50 = 50
Step 7. Answer the question. The length is 15 inches.

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Find the length of a rectangle with: perimeter 80 and width 25.

15

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Find the length of a rectangle with: perimeter 30 and width 6.

9

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We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for. the length and width of a rectangle
Step 3. Name. Choose a variable to represent it.
Since the width is defined in terms of the length, we let L = length. The width is two feet less than the length, so we let L − 2 = width.
.
P = 52 ft
Step 4. Translate .
Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. P = 2 L + 2 W
Substitute in the given information. 52 = 2 L + 2 ( L 2 )
Step 5. Solve the equation. 52 = 2 L + 2 L 4
Combine like terms. 52 = 4 L 4
Add 4 to each side. 56 = 4 L
Divide by 4. 56 4 = 4 L 4
14 = L
The length is 14 feet.
Now we need to find the width. The width is L 2 .
.
The width is 12 feet.
Step 6. Check.
Since 14 + 12 + 14 + 12 = 52 , this works!
.
Step 7. Answer the question. The length is 14 feet and the width is 12 feet.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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