10.8: Area of a Circle
Do you know the area of a circle is? Well to think about this, let's look once again at the small round table at Jillian's house.
In an earlier Concept, you learned how to measure around the edge of the circumference using the diameter or the radius. The diameter of this table is 4 feet.
Now, you are going to learn how to measure the flat top of the circle. The space inside the circumference of a circle is known as the area of the circle.
Given the diameter, what is the area of this table?
This Concept is all about calculating area. By the end of it, you will know how to solve this problem.
Guidance
In an earlier Concept, you learned about the parts of a circle and about finding the circumference of a circle. This Concept is going to focus on the inside of the circle. The inside of a circle is called the area of the circle.
What is the area of a circle?
Remember back to working with quadrilaterals? The area of the quadrilateral is the surface or space inside the quadrilateral. Well, the area of a circle is the same thing. It is the area inside the circle that we are measuring.
In this picture, the area of the circle is yellow. How do you measure the area of a circle?
To figure out the area of a circle, we are going to need a couple of different measurements. The first one is pi. We will need to use the numerical value for pi, or 3.14, to represent the ratio between the diameter and the circumference.
The next measurement we need to use is the radius. Remember that the radius is the distance from the center of a circle to the edge, or is 1/2 of the circle's diameter.
We calculate the area of a circle by multiplying the radius squared (multiplied by itself) by pi (3.14).
Here is the formula.
\begin{align*}A = \pi r^2\end{align*}
Find the area of the circle.
Next, we use our formula and the given information.
\begin{align*}A & = \pi r^2\\
A & = (3.14)(3^2)\\
A & = (3.14)(9)\\
A & = 28.26 \ mm^2\end{align*}
What about if you have been given the diameter and not the radius?
If you have been given the diameter and not the radius, you can still figure out the area of the circle. You start by dividing the measure of the diameter in half since the radius is onehalf the measure of the diameter.
Then you use the formula and solve for area.
Notice that the diameter is 6 inches. We can divide this in half to find the radius.
6 \begin{align*}\div\end{align*}
Next, we substitute the given values into the formula and solve for the area of the circle.
\begin{align*}A & = \pi r^2\\
A & = (3.14)(3^2)\\
A & = (3.14)(9)\\
A & = 28.26 \ in^2\end{align*}
Now it’s time for you to try a few on your own. Find the area of the circle using the given radius or diameter.
Example A
\begin{align*}r = 12 \ cm\end{align*}
Solution: 452.16 sq. cm.
Example B
\begin{align*}r = 18 \ cm\end{align*}
Solution: 1017.36 sq. cm
Example C
\begin{align*}d = 14 \ in\end{align*}
Solution: 153.86 sq. in.
Now let's go back to the original problem from the beginning of this Concept.
Do you know the area of a circle is? Well to think about this, let's look once again at the small round table at Jillian's house.
In an earlier Concept, you learned how to measure around the edge of the circumference using the diameter or the radius. The diameter of this table is 4 feet.
Now, you are going to learn how to measure the flat top of the circle. The space inside the circumference of a circle is known as the area of the circle.
Given the diameter, what is the area of this table?
If you have been given the diameter and not the radius, you can still figure out the area of the circle. You start by dividing the measure of the diameter in half since the radius is onehalf the measure of the diameter.
The diameter of the table is 4 feet, so the radius is 2 feet.
Then you use the formula and solve for area.
Next, we substitute the given values into the formula and solve for the area of the circle.
\begin{align*}A & = \pi r^2\\
A & = (3.14)(2^2)\\
A & = (3.14)(4)\\
A & = 12.56 \ ft^2\end{align*}
This is our answer.
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.
The diameter of a circle is 13 feet. What is the area of the circle?
Answer
To work on this problem, we must first divide the diameter in half to find the radius.
\begin{align*}13 \div 2 = 6.5\end{align*}
The radius is 6.5 feet.
Next, we substitute the given values into the formula for area and solve.
\begin{align*}A & = \pi r^2\\
A & = (3.14)(6.5^2)\\
A & = (3.14)(42.25)\\
A & = 132.665 \ ft^2\end{align*}
Next, we can round up the nearest hundredth.
The answer is 132.67 sq. feet.
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: Find the area of the following circles given the radius.
1. \begin{align*}r = 4 \ in\end{align*}
2. \begin{align*}r = 5 \ cm\end{align*}
3. \begin{align*}r = 8 \ in\end{align*}
4. \begin{align*}r = 2 \ cm\end{align*}
5. \begin{align*}r = 7 \ m\end{align*}
6. \begin{align*}r = 9 \ in\end{align*}
7. \begin{align*}r = 10 \ ft\end{align*}
8. \begin{align*}r = 11 \ cm\end{align*}
9. \begin{align*}r = 20 \ ft\end{align*}
10. \begin{align*}r = 30 \ miles\end{align*}
Directions: Find the area of the following circles given the diameter.
11. \begin{align*}d = 10 \ in\end{align*}
12. \begin{align*}d = 12 \ m\end{align*}
13. \begin{align*}d = 14 \ cm\end{align*}
14. \begin{align*}d = 16 \ ft\end{align*}
15. \begin{align*}d = 18 \ in\end{align*}
16. \begin{align*}d = 22 \ ft\end{align*}
17. \begin{align*}d = 24 \ cm\end{align*}
18. \begin{align*}d = 28 \ m\end{align*}
19. \begin{align*}d = 30 \ m\end{align*}
20. \begin{align*}d = 36 \ ft\end{align*}
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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 areas of circles given the radius or the diameter.