Anna wants to redecorate an old table. She wants to put decorative tape around the outside of the table. The problem is, she only knows the area of the table and one side length. She needs to know the other side length in order to get enough tape to cover all four sides. The area of the table is 32 square feet and the length of the table is 8 feet. Anna needs to solve for the missing dimension to determine if a roll of tape that is 25 feet long will be enough tape.

In this concept, you will learn how to figure out unknown dimensions of length or width when given the area or perimeter of a figure.

### Unknown Dimensions

**Dimensions** are measurements needed in order to find the area or perimeter of a square or rectangle. The dimensions that you are familiar with are length and width (or side length in a square). Sometimes, a math problem presents missing dimensions but provides the area or perimeter to help you solve the missing dimension.

For example, there is a square with a perimeter of 12 inches. Find the side length of the square.

First, write the formula for the perimeter of a square.

\begin{align*}P=4s\end{align*}

Next, fill in the known information. This problem provides the perimeter or \begin{align*}P\end{align*}

\begin{align*}12=4s\end{align*}

Then, to solve the equation, you can either rewrite the equation to make it a division problem, or ask yourself what number multiplied by 4 gives you a total of 12. Either strategy will get you the answer of 3.

\begin{align*}12\div 4&= s\\
3&=s\\
&or\\
12 &= 4(3)\\
12 &= 12\end{align*}

Sometimes a figure is presented and the area of that figure is provided.

For example, find the side length for a square with an area of 36 sq. in.

First, use the formula for finding the area of a square.

\begin{align*}A &= s \times s\\
36 &= s \times s\end{align*}

Then, think, “What number times itself will give me 36?” The answer is 6.

\begin{align*}36 &= 6 \times 6\\
36 &= 36\end{align*}

Double check your work by making sure the answer multiplied by itself gives the area. In this case, the answer checks out as 6 times 6 does equal 36.

This same concept works for rectangles, only you use the formula that is more appropriate for rectangles.

When the perimeter or area is given, you plug in the information given and solve for the missing dimension. The concept for area works the same as for a square only length and width are used instead of "s" for side length. When solving for a missing dimension when the perimeter is given in a rectangle, it looks a little different than with a square.

For example, find the width of a rectangle whose perimeter is 18 inches and the length is 6 inches.

First, write the formula and substitute the given information.

\begin{align*}18= 2(6) + 2(w)\end{align*}

This equation shows that the only variable present is the width, the missing dimension. Solve what can be solved first in the problem before isolating the variable "w".

Then, to isolate the 2w, subtract the 12 from both sides of the equation. That leaves the equation looking like this:

Now either rewrite the equation as division or ask yourself what number times 2 gives you 6.

### Examples

#### Example 1

Earlier, you were given a problem about Anna and her table.

She knows the area of the table is 32 sq. ft. and the length is 8 feet. She needs to find the width of the table to then determine if 25 ft. of tape is enough.

First, Anna writes out the formula to solve for the missing dimension.

Next, Anna plugs in the information she has already. She knows the area and the length.

Then, she solves for the missing dimensions (width) by dividing 32 by 8.

The missing width is 4 feet. But now, Anna needs to figure out if she has enough tape. Now that Anna knows both dimensions of the table, she can figure out the perimeter of the outside of the table by using the formula.

\begin{align*}P&= 2(8)+2(4)\\
P&=16+8\\
P&=24\end{align*}

The total perimeter of Anna's table is 24 feet, which means 25 feet of tape is enough.

#### Example 2

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

First, write the formula for area of a square.

Next, plug in the information given in the problem. The area is given so that substitutes for A

Then, figure out which number times itself will gives you 144. The answer is 12.

\begin{align*}144= 12 \times 12\end{align*}

The answer is 12 feet.

#### Example 3

Find the side length of a rectangle that has a perimeter of 48 feet and a width of 9 feet.

First, write out the formula for perimeter of a rectangle.

\begin{align*}P= 2(l)+2(w)\end{align*}

Next, plug in the given information. Solve that parts of the equation that can be solved already.

\begin{align*}48&= 2(l)+ 2(9)\\ 48&= 2(l)+ 18\end{align*}

Then, isolate the missing dimension variable and solve for the final answer.

\begin{align*}30&= 2(l)\\ 30\div 2&=l\\ 15&=l\end{align*}

The answer is 15 feet.

#### Example 4

Find the side length of a square that has a perimeter of 56 feet.

First, write the formula for perimeter of a square.

Next, plug in the given information.

\begin{align*}56= 4(s)\end{align*}

Then, solve for the side length.

The answer is 14 feet.

#### Example 5

Find the side length of a rectangle that has an area of 120 sq. miles and a length of 12 miles.

First, write the formula for area of a rectangle.

Next, plug in the given information.

Then, solve for the missing dimension.

\begin{align*}120\div 12&= w\\ 10&= w\end{align*}

The answer is 10 feet.

### Review

Find the side length of each square given its perimeter.

- P = 24 inches
- P = 36 inches
- P = 50 inches
- P = 88 centimeters
- P = 90 meters
- P = 20 feet
- P = 32 meters
- P = 48 feet

Find the side length of each square given its area.

- A = 64 sq. inches
- A = 49 sq. inches
- A = 121 sq. feet
- A = 144 sq. meters
- A = 169 sq. miles
- A = 25 sq. meters
- A = 81 sq. feet
- A = 100 sq. miles

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.6.