10.9: Radius or Diameter of a Circle Given Area
Remember Jillian and the tables for the quilters?
Well, Marie mentioned to Jillian that when she quilts at her home that she has a bigger table to work on. Marie said that the area of the table is 113.04 square feet.
Jillian thinks that Marie said it that way to make her think of the math.
Can you think of the math in this way? What would the diameter of this table be? What would the radius be?
In this Concept, you will learn how to work backwards to figure out the radius or diameter of a circle when given the area.
Guidance
Now that you know how to find the area of a circle given a radius or diameter, we can work backwards and use the area to find the radius or the diameter.
It is time to use your detective skills again!!
The area of the circle is \begin{align*}153.86 \ in^2\end{align*}
This problem asks for you to figure out two different things. First, let’s find the radius and then we can use that measure to figure out the diameter.
Let’s begin by using the formula for finding the area of a circle.
\begin{align*}A & = \pi r^ 2\\
153.86 & = (3.14)r^2\end{align*}
We substituted in the given information. We know the area, and we know that the measure for pi is 3.14. Next, we can divide the area by pi. This will help to get us one step closer to figuring out the radius.
\begin{align*}{3.14 \overline{ ) {153.86}}}\end{align*}
Remember, when you divide decimals, we move the decimal two places in the divisor and the dividend. Here is our new division problem.
\begin{align*}{314 \overline{ ) {15386}}}\end{align*}
Yes. It is a large number to divide, but don’t let that stop you. Just work through it step by step and you will be able to find the correct answer!
\begin{align*}& \overset{\qquad \ \quad 49}{314 \overline{ ) {15386}}}\\
& \quad \underline{1256 \ \ }\\
& \qquad \ 2826\\
& \quad \ \ \underline{2826}\\
& \qquad \qquad 0\end{align*}
So far, our answer is 49, but that is not the radius.
\begin{align*}49 = r^2\end{align*}
We need to figure out which number times itself is equal to 49.
Now we know that the radius is 7 because 7 \begin{align*}\times\end{align*}
The radius is 7 inches.
What is the diameter?
The measure of the radius is onehalf the measure of the diameter. If the radius is 7, then the diameter is double that.
7 \begin{align*}\times\end{align*}
The diameter is 14 inches.
Warning! Working backwards is tricky! Be sure that you take your time when working through problems!
Try a few of these on your own.
Example A
If the area of a circle is \begin{align*}12.56 \ cm^2\end{align*}
Solution: Radius = 2 cm, Diameter = 4 cm
Example B
If the area of a circle is \begin{align*}200.96 \ m^2\end{align*}
Solution: Radius = 8 m, Diameter = 16 m
Example C
If the area of a circle is \begin{align*}379.94 \ m^2\end{align*}
Solution: Radius = 11 m, Diameter = 22 m
Here is the original problem once again.
Remember Jillian and the tables for the quilters?
Well, Marie mentioned to Jillian that when she quilts at her home that she has a bigger table to work on. Marie said that the area of the table is 113.04 square feet.
Jillian thinks that Marie said it that way to make her think of the math.
Can you think of the math in this way? What would the diameter of this table be? What would the radius be?
Let’s begin by using the formula for finding the area of a circle.
\begin{align*}A & = \pi r^ 2\\
113.04 & = (3.14)r^2\end{align*}
Next, we divide 113.04 by 3.14.
\begin{align*}113.04 \div 3.14 = 36\end{align*}
Now we know we need to find the radius. We can figure this out by thinking "what times itself is equal to 36?"
Six!
The radius of the circular table is 6 feet.
The diameter of the circular table is 12 feet.
That is one big table!
Vocabulary
Here are the vocabulary words in this Concept.
 Area
 the surface or space of the figure inside the perimeter.
 Radius
 the measure of the distance halfway across a circle.
 Diameter
 the measure of the distance across a circle
 Squaring

uses the exponent 2 to show that a number is being multiplied by itself. \begin{align*}3^2 = 3 \times 3\end{align*}
32=3×3
 Pi
 the ratio of the diameter to the circumference. The numerical value of pi is 3.14.
Guided Practice
Here is one for you to try on your own.
If the area of a circle is 314 sq. cm, what is the radius of the circle?
Answer
\begin{align*}A & = \pi r^ 2\\
314 & = (3.14)r^2\end{align*}
Next, we can divide 314 by 3.14.
\begin{align*}100 = r^2\end{align*}
The radius of the circle is 10 cm.
Video Review
Here are videos for review.
Khan Academy, Area of a Circle
James Sousa, Example of Determining the Area of a Circle
Practice
Directions: Use each area to find the radius of each circle.
1. \begin{align*}A = 12.56\end{align*}
2. \begin{align*}A = 28.26\end{align*}
3. \begin{align*}A = 50.24\end{align*}
4. \begin{align*}A = 78.5\end{align*}
5. \begin{align*}A = 153.86\end{align*}
6. \begin{align*}A = 200.96\end{align*}
7. \begin{align*}A = 254.34\end{align*}
8. \begin{align*}A = 113.04\end{align*}
9. \begin{align*}A = 452.16\end{align*}
10. \begin{align*}A = 615.44\end{align*}
11. \begin{align*}A = 803.84\end{align*}
12. \begin{align*}A = 1017.36\end{align*}
13. \begin{align*}A = 1256\end{align*}
14. \begin{align*}A = 1384.74\end{align*}
15. \begin{align*}A = 1962.5\end{align*}
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Term  Definition 

(Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.  
Area  Area is the space within the perimeter of a twodimensional figure. 
Diameter  Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius. 
Pi  (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14. 
Radius  The radius of a circle is the distance from the center of the circle to the edge of the circle. 
Squaring  Squaring a number is multiplying the number by itself. The exponent 2 is used to show squaring. 
Image Attributions
Here you'll learn to find the radius or diameter of a circle when you have been given the area.