# 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?

### Area and Perimeter of Rectangles

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*}.

#### Calculating the Area and Perimeter

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*}.

#### Finding the Perimeter given the Area

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*}

#### Finding the Area and Perimeter

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*}.

#### Bed 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.

### Examples

#### Example 1

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

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

#### Example 2

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

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*}.

#### Example 3

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

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

### Review

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?

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### Vocabulary Language: English

TermDefinition
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.