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Unknown Dimensions Using Formulas

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Unknown Dimensions Using Formulas

What about a larger garden? Alex is wondering how perimeter and area are affected if the garden is larger. Will the same formulas from the last two Concepts work? What if you were given the area and needed to figure out a side length? Could you do it? Take a look at this dilemma.

On Sunday, Alex visited a botanical garden. Because he had been working on his own garden plot, Alex noticed the garden beds and their designs in new ways. One garden was so beautiful that he stopped to read about it.The plot was a square plot and was full of beautiful flowers. The sign said that the area of the plot was 484 square feet.

If this is the area of the plot, what is the side length? What is the perimeter of the plot?

In this Concept, you will learn how to figure out unknown dimensions. Then you can answer these two questions at the end of the Concept.

Guidance

The side length of a square or the length and width of a rectangle can be called the dimensions or the measurements of the figure. We just finished figuring out the area and perimeter of squares and rectangles when we were given the dimensions of the figure. Can we do this work backwards?

Can we figure out the dimensions of a square when we have been given the perimeter or area of the square?

Hmmmm. This is a bit tricky. We will still need to use the formula, but we will need to “think backwards” in a way. If the perimeter of the square is 12 inches, what is the side length of the square?

To complete this problem, we are going to need to work backwards. Let’s start by using the formula for the perimeter of a square.

P=4s

Next, we fill in the information that we know. We know the perimeter or P .

12=4s

We can ask ourselves, “What number times four will give us 12?” The answer is 3. We can check our work by substituting 3 in for s to see if we have a true statement.

12 &= 4(3)\\12 &= 12

Our answer checks out.

Now let’s look at how we can figure out the side length of a square when we have been given the area of the square.

Area = 36 sq. in.

We know that the area of the square is 36 square inches. Let’s use the formula for finding the area of a square to help us.

A &= s \times s\\36 &= s \times s

We can ask ourselves, “What number times itself will give us 36?” Our answer is 6. Because we have square inches, we know that our answer is 6 inches. We can check our work by substituting 6 into the formula for finding the area of a square.

36 &= 6 \times 6\\36 &= 36 Our answer checks out.

Now let's practice with a few examples.

Example A

What is the side length of a square that has a perimeter of 48 feet?

Solution: 12 feet

Example B

What is the side length of a square that has a perimeter of 56 feet?

Solution: 14 feet

Example C

What is the side length of a square that has an area of 121 sq. miles?

Solution: 11 miles

Now let's think back to the garden that Alex saw at the Botanical Garden. Here is the problem once again.

On Sunday, Alex visited a botanical garden. Because he had been working on his own garden plot, Alex noticed the garden beds and their designs in new ways. One garden was so beautiful that he stopped to read about it.The plot was a square plot and was full of beautiful flowers. The sign said that the area of the plot was 484 square feet.

If this is the area of the plot, what is the side length? What is the perimeter of the plot?

We know that the garden plot is square, so let's start with area. We need to figure out what number times what number is equal to 484. To do this, we can use guess and check. We know that 20 times 20 equals 400. Therefore, let's try a number a little larger than 20.

22 \times 22 = 484

The side length of the square plot is 22 feet.

Now we can go back to perimeter. If the side length of the square plot is 22 feet, then we can multiply that number by 4 and get the total perimeter.

22 \times 4 = 88

The perimeter of the square plot is 88 feet.

Vocabulary

Perimeter
the distance around the edge of a figure.
Square
a figure with four congruent sides
Formula
a way or method of solving a problem
Rectangle
a figure that has opposite sides that are congruent
Area
the space inside the edges of a figure
Dimensions
the measurements that define a figure

Guided Practice

Here is one for you to try on your own.

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

Answer

First, we have to figure out which number times itself will give us 144. The answer is 12.

12 \times 12 = 144

The side length of the square is 12 feet.

Now we can figure out the perimeter by multiplying 12 times 4.

12 \times 4 = 48

The perimeter of the square is 48 feet.

Video Review

Khan Academy Area and Perimeter

James Sousa Area and Perimeter

James Sousa An Example of Area and Perimeter

Practice

Directions: Find the side length of each square given its perimeter.

1. P = 24 inches

2. P = 36 inches

3. P = 50 inches

4. P = 88 centimeters

5. P = 90 meters

6. P = 20 feet

7. P = 32 meters

8. P = 48 feet

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

9. A = 64 sq. inches

10. A = 49 sq. inches

11. A = 121 sq. feet

12. A = 144 sq. meters

13. A = 169 sq. miles

14. A = 25 sq. meters

15. A = 81 sq. feet

16. A = 100 sq. miles

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