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# Area of Squares and Rectangles

## Find total surface measures of rectilinear quadrilaterals.

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Area of Squares and Rectangles

### Let's Think About It

License: CC BY-NC 3.0

Allen bought a new house and is planning to replace the carpet in two of the bedrooms. One bedroom is a shaped like a square and the other is rectangular. He needs to figure out how much carpet is needed to cover both bedroom floors.

The square bedroom has side lengths of 9 feet.

The rectangular bedroom has a length of 12 feet and a width of 8 feet.

In order for Allen to figure out how much carpet he needs to buy for these two bedrooms, he needs to find the area of each bedroom.

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

### Guidance

The inside space of a figure is known as the area of the figure. Carpeting a floor, grass on the ground, or anything else covering the space inside of a figure are all examples of area.

There is a formula for finding the area of different shapes. In this concept, this concept will show you how to find the area of both squares and rectangles.

This is the formula used to find the area of a square:

\begin{align*}A= s\cdot s\end{align*}

Remember, the dot is another symbol for multiplication. To figure out the area of the square, multiply one side times the other side. Since all sides in a square are equal, the two numbers will be the same. The unit of measurement is kept in the equation to help you remember what the correct label should be.

Here is how to use the formula for finding the area of a square:

\begin{align*}A=6 \ ft \cdot 6 \ ft\end{align*}

Now, think of the equation as multiplying two different things; the numbers and the units of measurement.

\begin{align*}A &= 6 \cdot 6\\ A &= ft \cdot ft\end{align*}

Solve both equations.

\begin{align*}A &= 6 \times 6 = 36\\ A &= ft \times ft = ft^2\end{align*}

Think back to what you know about exponents. When you multiply two of the same thing together, you can write it in exponential form. Another way to label area is to write square feet (sq. ft.).

The answer is 36 sq. ft. or \begin{align*}36 {ft}^{2}\end{align*}

This is the formula used to find the area of a rectangle:

\begin{align*}A= lw\end{align*}

To find the area of a rectangle, use the measurements for length and width instead of side length. These numbers may be different from each other because not all rectangles have the same side lengths like squares do.

This rectangle measures a length of 5 meters and a width of 3 meters. Just like the area formula for a square, multiply those two numbers to find the area of the rectangle. Remember, two variables next to each other in an equation indicate multiplication, just like the dot symbol or traditional multiplication symbol. The symbols will be used interchangeably.

To find the area of a rectangle, multiply length times width.

\begin{align*}A &= 5m \cdot 3m\\ A &= 5 \times 3\\ A &= meters \times meters\\\end{align*}

The formula shows 5 meters times 3 meters. Multiply the measurement part (5 \begin{align*}\times\end{align*} 3), then multiply the units of measurement.

The answer is 15 sq. m or \begin{align*}15 \ m^2\end{align*}.

The label for area of a rectangle is similar to the label for area of a square from above. The label can be written in exponential form or written in abbreviation.

### Guided Practice

Find the area of a rectangle with a length of 7 ft. and a width of 4 ft.

First, write out the equation

\begin{align*}A&=lw \\ A&=7ft \cdot 4ft\end{align*}

Next, multiply the two numbers.

\begin{align*}A= 7 \cdot 4 = 28\end{align*}

Then, multiply the units of measurement.

\begin{align*}A= ft \cdot ft = {ft}^{2}\end{align*}

Now, put your answer together to show the final solution.

\begin{align*}A= 28 {ft}^{2}\end{align*}

The answer is 28 sq. ft.

Once you do more examples you may find yourself skipping the step of multiplying the units of measurement. This may become natural to square the unit of measurement for any area problem.

### Examples

#### Example 1

Find the area of a square with side lengths of 10 inches.

First, write your equation.

\begin{align*}A&= s \cdot s \\ A&= 10 in \cdot 10 in\end{align*}

Next, solve the equation by multiplying the two side lengths.

\begin{align*}A= 10 \cdot 10 = 100\end{align*}

Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.

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

The answer is 100 sq. in.

#### Example 2

Find the area of a rectangle with a length of 6 feet and a width of 5 feet.

First, write your equation.

\begin{align*}A&= lw \\ A&= 6ft \cdot 5ft\end{align*}

Next, solve the equation by multiplying the length times the width.

\begin{align*}A= 6 \cdot 5 = 30\end{align*}

Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.

\begin{align*}A= 30 {ft}^{2}\end{align*}

The answer is 30 sq. ft.

#### Example 3

Find the area of a rectangle with a length of 9 meters and a width of 8 meters.

First, write your equation.

\begin{align*}A&= lw \\ A&= 9m \cdot 8 m\end{align*}

Next, solve the equation by multiplying the length times the width.

\begin{align*}A= 9 \cdot 8 = 72\end{align*}

Then, write your final solution. Remember to write the squared unit of measurement after the numerical answer.

\begin{align*}A= 72 {m}^{2}\end{align*}

The answer is 72 sq. m

License: CC BY-NC 3.0

Remember Allen and his new house that he wants to re-carpet?

The dimensions of Allen's two bedrooms are given in the beginning of the concept. His square bedroom has side lengths of 9 feet. The other room is a rectangle with dimensions of 12 feet by 8 feet. To begin solving this problem, find the area of each of the bedrooms.

The square bedroom has a side length of 9 feet.

\begin{align*}A &= s \cdot s\\ A &= 9 \times 9 = 81 \ sq. \ feet\end{align*}

The square bedroom has an area of 81 square feet.

The rectangular bedroom has a length of 12 feet and width of 8 feet.

\begin{align*}A &= 12ft \cdot 8ft\\ A &= 12 \times 8\\ A &= feet \times feet\end{align*}

The rectangular bedroom has an area of 96 square feet.

Allen wants the total area of his two bedrooms to know how much carpet to buy, so add the two areas together.

\begin{align*}81 + 96 = 177 \ square \ feet\end{align*}

The answer is 177 square feet.

### Explore More

Find the area of each of the following figures. Make sure you label your answer correctly.

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 13 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.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.5.

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### Vocabulary Language: English

TermDefinition
Area Area is the space within the perimeter of a two-dimensional figure.
Dimensions Dimensions are the measurements that define the shape and size of a figure.
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.