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Radius or Diameter of a Circle Given Area

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Radius or Diameter of a Circle Given Area

Remember Terrence and his swimming in the Area of a Circle Concept? Take a look at this problem.

Terrence swam in a pool with an area of 452.16 sq. feet. What is the diameter of this pool?

To figure this out, you will need to know how to problem solve to find the radius or diameter of a circle. This Concept will teach you how to do this.

Guidance

We have seen that when we are given the radius or the diameter of a circle, we can find its area. We can also use the formula to find the radius or diameter if we know the area. Let’s see how this works.

The area of a circle is 113.04 square inches. What is its radius?

This time we know the area and we need to find the radius. We can put the number for area into the formula and use it to solve for the radius, $r$ .

$A & = \pi r^2\\113.04 & = \pi r^2\\113.04 \div \pi & = r^2\\36 & = r^2\\\sqrt{36} & = r \\6 \ in. & = r$

Let’s look at what we did to solve this. To solve this problem we needed to isolate the variable $r$ . First, we divided both sides by $\pi$ , or 3.14. Then, to remove the exponent, we took the square root of both sides. A square root is a number that, when multiplied by itself, gives the number shown. We know that 6 is the square root of 36 because $6 \times 6 = 36$ .

The radius of a circle with an area of 113.04 square inches is 6 inches.

What is the diameter of a circle whose area is $379.94 \ cm^2$ ?

What is this problem asking us to find? We need to find the diameter (not the radius!). What information is given in the problem? We know the area. Therefore we can use the formula to solve for the radius, $r$ . Once we know the radius, we can find the diameter. Let’s give it a try.

$A & = \pi r^2\\379.94 & = \pi r^2\\379.94 \div \pi & = r^2\\121 & = r^2\\\sqrt{121} & = r\\11 \ cm & = r$

The radius of a circle with an area of 379.94 square centimeters is 11 centimeters.

Remember, this problem asked us to find the diameter, so we’re not done yet. How can we find the diameter?

The diameter is always twice the length of the radius, so the diameter of this circle is $11 \times 2 = 22$ centimeters.

As we have seen, we can use the area formula whenever we are given information about a circle. If we know the diameter or radius, we can solve for the area, $A$ . If we are given the area, we can solve for the radius, $r$ . If we know the radius, we can also find the diameter.

Find the radius of each circle.

Example A

Area = 153.86 sq. in.

Solution: $7$ inches

Example B

Area = 379.94 sq. ft.

Solution: $11$ ft.

Example C

452.16 sq. m

Solution: $12$ meters

Here is the original problem once again.

Terrence swam in a pool with an area of 452.16 sq. feet. What is the diameter of this pool?

To figure this out, we need to use the formula for area and then solve using the given information.

$A & = \pi r^2\\452.16 & = \pi r^2\\452.16 \div \pi & = r^2\\144 & = r^2\\\sqrt{169} & = r \\12 \ ft. & = r$

We need to find the diameter of the pool. We can figure this out by doubling the radius.

The diameter of the pool is 24 feet.

Vocabulary

Circle
a set of connected points that are equidistant from a center point.
Diameter
the distance across the center of a circle.
the distance from the center of the circle to the outer edge.
Area
the space inside a two-dimensional figure.

Guided Practice

Here is one for you to try on your own.

What is the radius of a circle with an area of 530.66 sq. inches?

To figure this out, we use the formula for finding the area of circle, then we solve using our given information.

$A & = \pi r^2\\530.66 & = \pi r^2\\530.66 \div \pi & = r^2\\169 & = r^2\\\sqrt{169} & = r \\13 \ in. & = r$

The radius of this circle is 13 inches.

Practice

Directions: Find each radius given the area of the circle.

1. 12.56 sq. in.

2. 78.5 sq. m

3. 200.96 sq. cm

4. 254.34 sq. in

5. 7.07 sq. ft.

6. 28.26 sq. m

Directions: Find each diameter given the area of the circle.

7. 12.56 sq. in.

8. 78.5 sq. m

9. 200.96 sq. cm

10. 254.34 sq. in

11. 7.07 sq. ft.

12. 28.26 sq. m

13. 78.5 sq. feet

14. 176.625 sq. m

15. 113.04 sq. ft.