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Area of a Circle

Pi times the radius squared.

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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*}A=πr2

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*}AAAA=πr2=(3.14)(32)=(3.14)(9)=28.26 mm2

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 one-half 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*}÷ 2 \begin{align*}=\end{align*}= 3 inches \begin{align*}=\end{align*}= radius

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*}AAAA=πr2=(3.14)(32)=(3.14)(9)=28.26 in2

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*}r=12 cm

Solution: 452.16 sq. cm.

Example B

\begin{align*}r = 18 \ cm\end{align*}r=18 cm

Solution: 1017.36 sq. cm

Example C

\begin{align*}d = 14 \ in\end{align*}d=14 in

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 one-half 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*}AAAA=πr2=(3.14)(22)=(3.14)(4)=12.56 ft2

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*}13÷2=6.5

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*}AAAA=πr2=(3.14)(6.52)=(3.14)(42.25)=132.665 ft2

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*}r=4 in

2. \begin{align*}r = 5 \ cm\end{align*}r=5 cm

3. \begin{align*}r = 8 \ in\end{align*}r=8 in

4. \begin{align*}r = 2 \ cm\end{align*}r=2 cm

5. \begin{align*}r = 7 \ m\end{align*}r=7 m

6. \begin{align*}r = 9 \ in\end{align*}r=9 in

7. \begin{align*}r = 10 \ ft\end{align*}r=10 ft

8. \begin{align*}r = 11 \ cm\end{align*}r=11 cm

9. \begin{align*}r = 20 \ ft\end{align*}r=20 ft

10. \begin{align*}r = 30 \ miles\end{align*}r=30 miles

Directions: Find the area of the following circles given the diameter.

11. \begin{align*}d = 10 \ in\end{align*}d=10 in

12. \begin{align*}d = 12 \ m\end{align*}d=12 m

13. \begin{align*}d = 14 \ cm\end{align*}d=14 cm

14. \begin{align*}d = 16 \ ft\end{align*}d=16 ft

15. \begin{align*}d = 18 \ in\end{align*}d=18 in

16. \begin{align*}d = 22 \ ft\end{align*}d=22 ft

17. \begin{align*}d = 24 \ cm\end{align*}d=24 cm

18. \begin{align*}d = 28 \ m\end{align*}d=28 m

19. \begin{align*}d = 30 \ m\end{align*}d=30 m

20. \begin{align*}d = 36 \ ft\end{align*}d=36 ft

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Vocabulary

\pi

\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.

Area

Area is the space within the perimeter of a two-dimensional 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 (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.

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