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

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Kelly is faced with the following dilemma.

The area of a rectangle is 240 square feet. The length of one side is 15 feet. Write and solve an equation to determine the width of one side of the rectangle.

Do you know how to do this? Pay attention to this Concept and you will learn all about area and perimeter.

Guidance

This Concept is all about formulas. Let’s start by thinking about the definition of a formula.

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

By this point in mathematics, there are many formulas that you already have used to solve problems. Now let's explore some of those familiar formulas. Let’s start 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 of a rectangle is: $2L + 2W$ . $L =$ length and $W =$ width.

We can solve for the perimeter of this rectangle by substituting the given values for the variables that represent length and width.

$2(9) + 2(12) &= \text{Perimeter}\\18 + 24 &= \text{Perimeter}\\42 \ inches &= \text{Perimeter}$

The perimeter of this rectangle is 42 inches.

Area is the amount of square units inside the figure. Area is found by multiplying the $\text{Length} \times \text{Width}$ . The formula for finding the area of a rectangle is $L \times W$ .

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

$12 \times 9 &= \text{Area}\\12 \times 9 &= \text{Area}\\108 \ cm^2 &= \text{Area}$

The area of this rectangle is $108 \ cm^2$ .

Copy these two formulas and the note about square units in your notebook. Use a drawing if necessary to help you remember each formula.

We can also find the perimeter and area of a square. Remember that a square has four equal sides, so we can use the following formula for finding the perimeter of a square, $4s$ .

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.

Recall that the formula to determine perimeter of a rectangle is $2(L) + 2(W)$ . Substitute the information given in the problem to determine the missing dimension. We know the length of one side is 12 feet. Therefore, plug in 12 for the length. We also know that the total perimeter is 72 feet. Therefore, set the formula equal to 72 feet. Then, use inverse operations to solve for the unknown width.

$\text{Perimeter} &= 2(L) + 2(W)\\\text{Perimeter} &= 2(12)+ 2(W)= 72\\\text{Perimeter} &= 24 + 2W = 72$

Next, we need to work to figure out the value of the width. It makes sense to use an inverse operation and subtract 24 from 72.

$2W=48$

Two times twenty-four is forty-eight. This is our answer.

The unknown width is 24 feet.

Example A

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

Solution:   The perimeter of the square is 18 inches.

Example B

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

Solution:   The area of the square is $20.25 \ in^2$ .

Example C

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

Solution: $12$ meters

Now let's go back to the dilemma at the beginning of the Concept.

Recall that the formula for area is $L \times W$ . Plug in the information given in the problem. Then, use algebra to solve for the unknown width.

$L \times W &= Area\\15W &= 240$

To figure this out, we divide 240 by 15. This is an example of using an inverse operation.

$W = 16 \ feet$

We can check our work by substituting 16 for the width in the equation $L \times W = Area$ .

$15 \times 16 &= \text{Area}\\15 \times 16 &= 240 \ square \ feet$

The rectangle is 15 feet by 16 feet.

Vocabulary

Formula
a method proven to work when solving specific types of problems.
Perimeter
the distance around a figure.
Area
the measurement of the inside of a figure.

Guided Practice

Here is one for you to try on your own.

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

Solution

Recall that a square has four equal sides. Therefore, $4s = 196 \ inches$ . Use inverse operations to solve for “ $s$ .”

$\frac{4s}{4} &= 196\\s &= 49 \ inches$

You can check your work by substituting 49 for the variable in $4s$ . 49 is the correct length if your answer is 196 inches.

$4s &= 196\\4(49) &= 196\\196 &= 196$

The length of one side of the square is 49 inches.

Explore More

Directions: 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.

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

1. $A = 64 \ in^2$
2. $A = 36 \ in^2$
3. $A = 81 \ m^2$
4. $A = 100 \ in^2$
5. $A = 144 \ ft^2$
6. $A = 121 \ cm^2$
7. $A = 4 \ mm^2$

Basic

Date Created:

Dec 19, 2012

Aug 21, 2014
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