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# Area and Perimeter of Rectangles

## Area is base times height, while perimeter is the sum of the sides.

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Area and Perimeter of Rectangles

What if Ed’s parents were getting him a new bed, and he had to decide what size bed is best for him? Initially he decided that he would like a king bed. Upon further research, Ed discovered there are two types of king beds, an Eastern (or standard) King and a California King. The Eastern King has \begin{align*}76'' \times 80''\end{align*} dimensions, while the California King is \begin{align*}72'' \times 84''\end{align*} (both dimensions are width \begin{align*}\times\end{align*} length). Which bed has a larger area to lie on? Which one has a larger perimeter? If Ed is 6’4”, which bed makes more sense for him to buy? After completing this Concept, you'll be able to use your knowledge of rectangles to answer these questions.

### Guidance

To find the area of a rectangle calculate the product of its base (width) and height (length) \begin{align*}A=bh\end{align*}. The perimeter of a rectangle is \begin{align*}P=2b+2h\end{align*}, where \begin{align*}b\end{align*} is the base (or width) and \begin{align*}h\end{align*} is the height (or length). If a rectangle is a square, with sides of length \begin{align*}s\end{align*}, the formula for perimeter is \begin{align*}P_{square}=2s+2s=4s\end{align*} and the formula for area is \begin{align*}A_{square}=s \cdot s=s^2\end{align*}.

#### Example A

Find the area and perimeter of a rectangle with sides 4 cm by 9 cm.

The perimeter is \begin{align*}4 + 9 + 4 + 9 = 26 \ cm\end{align*}. The area is \begin{align*}A=9 \cdot 4=36 \ cm^2\end{align*}.

#### Example B

The area of a square is \begin{align*}75 \ in^2\end{align*}. Find the perimeter.

To find the perimeter, we need to find the length of the sides.

\begin{align*}A = s^2 & =75 \ in^2\\ s &= \sqrt{75}=5 \sqrt{3} \ in\\ \text{From this}, \ P& =4 \left( 5 \sqrt{3} \right)=20 \sqrt{3} \ in.\end{align*}

#### Example C

Find the area and perimeter of a rectangle with sides 13 m and 12 m.

The perimeter is \begin{align*}2(13)+2(12)=50 \ m\end{align*}. The area is \begin{align*}13(12)=156 \ m^2\end{align*}.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

The area of an Eastern King is \begin{align*}6080 \ in^2\end{align*} and the California King is \begin{align*}6048 \ in^2\end{align*}. The perimeter of both beds is 312 in. Because Ed is 6’4”, he should probably get the California King because it is 4 inches longer.

### Vocabulary

Perimeter is the distance around a shape. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.” Area is the amount of space inside a figure. Area is measured in square units.

### Guided Practice

1. Find the area and perimeter of a square with side 5 in.

2. Draw two different rectangles with an area of \begin{align*}36 \ cm^2\end{align*}.

3. Find the area and perimeter of a rectangle with sides 7 in and 10 in.

1. The perimeter is \begin{align*}4(5)=20in\end{align*} and the area is \begin{align*}5^2=25 \ in^2\end{align*}.

2. Think of all the different factors of 36. These can all be dimensions of the different rectangles.

Other possibilities could be \begin{align*}6 \times 6, 2 \times 18\end{align*}, and \begin{align*}1 \times 36\end{align*}.

3. Area is \begin{align*}7(10)=70 \ in^2\end{align*}. Perimeter is \begin{align*}2(7)+2(10)=34 \ in\end{align*}.

### Practice

1. Find the area and perimeter of a square with sides of length 12 in.
2. Find the area and perimeter of a rectangle with height of 9 cm and base of 16 cm.
3. Find the area and perimeter of a rectangle if the height is 8 and the base is 14.
4. Find the area and perimeter of a square if the sides are 18 ft.
5. If the area of a square is \begin{align*}81 \ ft^2\end{align*}, find the perimeter.
6. If the perimeter of a square is 24 in, find the area.
7. The perimeter of a rectangle is 32. Find two different dimensions that the rectangle could be.
8. Draw two different rectangles that haven an area of \begin{align*}90 \ mm^2\end{align*}.
9. True or false: For a rectangle, the bigger the perimeter, the bigger the area.
10. Find the perimeter and area of a rectangle with sides 17 in and 21 in.

For problems 11 and 12 find the dimensions of the rectangles with the given information.

1. A rectangle with a perimeter of 20 units and an area of \begin{align*}24 \ units^2\end{align*}.
2. A rectangle with a perimeter of 72 units and an area of \begin{align*}288 \ units^2\end{align*}.
3. A rectangle with perimeter 138 units is divided into 8 congruent rectangles as shown in the diagram below. Find the perimeter and area of one of the 8 congruent rectangles.
4. The length of a rectangle is 2 more than 3 times the width. The perimeter of the rectangle is 44 units. What is the area of the rectangle?
5. The length of a rectangle is 2 less than 2 times the width. The area of the rectangle is \begin{align*}84 \ units^2\end{align*}. What is the perimeter of the rectangle?

### Vocabulary Language: English

Area of a Rectangle

To find the area 'A' of a rectangle, calculate A = bh, where b is the base (width) and h is the height (length).

Perimeter of a Rectangle

The perimeter 'P' of a rectangle is equal to twice the base added to twice the height: P = 2b + 2h.