Have you ever wondered about dimensions of a park or garden plot? Take a look at this dilemma.

Carmen and Jose have a beautiful spot in their yard for a garden. They hope to grow all kinds of vegetables and hopefully plant blueberry bushes along the outside edge of the garden. The square plot is perfect and the area of the plot is 169 square feet.

Given this information, can you figure out the side lengths of the garden?

**This Concept will teach you how to take an area or a perimeter and figure out missing dimensions. You will be using formulas to do this.**

### Guidance

Previously we worked on how to find the perimeter and area given measurements, now we are going to work backwards.

An ** equation** states that two expressions are equal and that a

**is an equation that includes an algebraic unknown, or a variable. Our formulas for perimeter and area are equations. In the examples above, the perimeter \begin{align*}(P)\end{align*} and the area \begin{align*}(A)\end{align*} were the variables. In some cases, if we already know the perimeter or the area, we can make the length or width our variable and solve for that value.**

*variable equation*The width of a rectangle is 10 feet and the perimeter is 50 feet. What is the length of the rectangle?

Here we are trying to find the length of the rectangle, so we’ll give that value the variable \begin{align*}l\end{align*}. We already know the perimeter and the width. All we have to do is substitute the values for the perimeter and the width into our formula and solve for \begin{align*}l\end{align*}.

*Remember to follow the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.*

\begin{align*}P &= 2l + 2w\\ 50 &= 2l + 2(10)\\ 50 &= 2l + 20\end{align*}

We can use mental math to find a number plus 20 that will give us 50. 30 is a likely choice. Now we have \begin{align*}2l\end{align*} so that means 2 times some number. We want this product to be 30, so \begin{align*}l=15\end{align*}.

Let’s check our work by substituting 15 in for length. If both sides of the equation are equal, then our equation balances and our work is done.

\begin{align*}P &= 2l+2w\\ 50 &= 2(15)+ 2(10)\\ 50 &= 30+20\\ 50 &= 50\end{align*}

Our equation balances and our work is done.

What about finding the side length when you have the area of a square?

If the area of a square is 144 sq. miles, what is the length of one of the sides of the square?

To do this, we work backwards and substitute the given area into the formula.

\begin{align*}A &= s^2\\ 144 &= s^2\end{align*}

Now we use mental math and figure out what number times itself is equal to 144? The answer is 12.

**The side length is 12 miles.**

**We can check our work by substituting the given values into the formula.**

\begin{align*}144 &= 12^2\\ 144 &= 144\end{align*}

**Our equation balances and our work is accurate.**

Now you can apply what you have learned to real-world problems.

The perimeter of a square playground is 200 yards. What is the length of one of the sides of the playground?

**This is one of those working backwards problems. Fill the given information into the formula and solve for the missing side length.**

\begin{align*}P &= 4s\\ 200 &= 4s\end{align*}

**What number times four is equal to 200? 50 is the correct answer. \begin{align*}50 \times 4 = 200\end{align*}**

**The side length of one side of the playground is 50 yards.**

Now it's time for you to try a few on your own.

#### Example A

If the perimeter of a figure is 30 feet, with a length of 9 feet, what is the width of the figure?

**Solution: \begin{align*}6 \ ft.\end{align*}**

#### Example B

If the area of a square is 225 sq. feet, what is the length of one of its sides?

**Solution: \begin{align*}15 \ ft.\end{align*}**

#### Example C

If the perimeter of a square is 24 meters, what is the length of one of the sides?

**Solution: \begin{align*}6 \ meters\end{align*}**

Here is the original problem once again.

Carmen and Jose have a beautiful spot in their yard for a garden. They hope to grow all kinds of vegetables and hopefully plant blueberry bushes along the outside edge of the garden. The square plot is perfect and the area of the plot is 169 square feet.

Given this information, can you figure out the side lengths of the garden?

To start with, let's look at what we know.

We know the area of the garden plot.

\begin{align*}169 \ sq. ft\end{align*}

We know that the garden plot is square.

Let's use the formula for area of a square to find the missing side.

\begin{align*}A = s^2\end{align*}

\begin{align*}169 = s^2\end{align*}

Next, we can ask ourselves, what value times itself is 169?

The answer is 13.

**The length of the side of the garden is \begin{align*}13\end{align*} feet.**

### Vocabulary

- Measurement
- the system of comparing an object to a standard.

- Perimeter
- the distance around the edge of a figure.

- Area
- the measurement inside a figure.

### Guided Practice

Here is one for you to try on your own.

The Hegazzi family is designing a summer garden based on the model shown below. Plot \begin{align*}A\end{align*} is a square and plot \begin{align*}B\end{align*} is a rectangle. If the total area of both plots in the garden is 139 square feet, what is the length of one side of plot \begin{align*}A\end{align*}?

**Answer**

To solve this multi-step problem, we will have to find the area of plot \begin{align*}B\end{align*} and subtract it from the total area to find the area of plot \begin{align*}A\end{align*}. Then, since we know plot \begin{align*}A\end{align*} is a square, we will have to determine what length *multiplied by itself* will give us the length of one side of plot \begin{align*}A\end{align*}.

Let’s get started. Plot \begin{align*}B\end{align*} is a rectangle. The formula for the area of a rectangle is \begin{align*}A = lw\end{align*}.

\begin{align*}A &= lw\\ A &= 15(6)\\ A &= 90 \ ft^2\end{align*}

So the area of plot \begin{align*}B\end{align*} is \begin{align*}90 \ ft^2\end{align*}. We know the total area of the garden is \begin{align*}139 \ ft^2\end{align*}, but we need to find the area of plot \begin{align*}A\end{align*}.

We can use the following equation: Total area = area of plot \begin{align*}A\end{align*} + area of plot \begin{align*}B\end{align*}. Let’s assign the value of the area of plot \begin{align*}A\end{align*} the variable \begin{align*}x\end{align*}.

Total area = area of plot \begin{align*}A\end{align*} + area of plot \begin{align*}B\end{align*}.

\begin{align*}139 = x + 90\end{align*}

“What number plus 90 is 139?” We can figure this out by thinking about subtracting 90 from 139. The answer is 49.

\begin{align*}x=49\end{align*}

Now we know the area of plot \begin{align*}A = 49 \ ft^2\end{align*}, but we are looking for the length of one side. **Plot \begin{align*}A\end{align*} is a square and all the sides are equal. The formula for the area of a square is \begin{align*}A = s^2\end{align*}. Let’s see how it looks:**

\begin{align*}A &= s^2\\ 49 &= s^2\end{align*}

We need to think of what number *times itself* is equal to 49. \begin{align*}7 \times 7 = 49\end{align*}, so the length of one side of plot \begin{align*}A\end{align*} is 7 ft.

### Video Review

This is a James Sousa video on how to find the perimeter of a given rectangle.

This is a James Sousa video on how to determine the area of a given rectangle.

### Practice

Directions: Given the area, find the side length for each square.

1. A = 16 sq. ft.

2. A = 64 sq. m.

3. A = 100 sq. miles

4. A = 121 sq. inches

5. A = 144 sq. feet

Directions: Given the area and length, find the width of each rectangle.

6. A = 24 sq. feet, length = 8 feet

7. A = 48 sq. feet, length = 12 feet

8. A = 64 sq. feet, length = 10 feet

9. A = 120 sq. meters, length = 40 meters

10. A = 130 sq. feet, length = 13 feet

11. A = 90 sq. inches, length = 45 inches

Directions: Given the perimeter, find the side length of each square.

12. P = 48 inches

13. P = 64 inches

14. P = 90 inches

15. P = 35 feet