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# Parts of Circles

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Circles

How does the area of a circle relate to the area of a rectangle?

#### Watch This

http://www.youtube.com/watch?v=SIKkWLqt2mQ James Sousa: Determine the Area of a Circle

http://www.youtube.com/watch?v=sHtsnC2Mgnk James Sousa: Determine the Circumference of a Circle

#### Guidance

A circle is the set of all points equidistant from a center point. From the center point to the circle is called the radius $(r)$ . From one side of the circle to the other through the center point is called the diameter $(d)$ .

The perimeter of a circle is called its circumference . The ratio between the circumference and diameter of any circle is $\pi$ . $\pi$ , or “pi”, is a Greek letter that stands for an irrational number approximately equal to 3.14. Because  $\pi$ is the ratio between the circumference and the diameter, the circumference of a circle is equal to the diameter times $\pi$ .

$C=\pi d$

The area of a circle is also related to $\pi$ .

$A=\pi r^2$

Example A

Find the area and circumference of a circle with radius 5 inches.

Solution:

$Area =\pi r^2=\pi(5^2)=25 \pi \ in^2$

$Circumference =\pi d=\pi(2.5)=10 \pi \ in$

Leaving your answer with a  $\pi$ symbol in it is called leaving your answer “in terms of $\pi$ ”. This is often preferable because it is the exact answer. As soon as you approximate a value for  $\pi$ your answer is not exact. Keep in mind that  $\pi$ is not the units. You should still put the appropriate units on your answers.

Example B

Find the area and circumference of a circle with diameter 16 cm.

Solution: If the diameter is 16 cm, then the radius is 8 cm.

$Area =\pi r^2=\pi(8^2)=64 \pi \ cm^2$

$Circumference =\pi d=16 \pi \ cm$

Example C

The shape below is a portion of a circle called a sector. This sector is  $\frac{1}{4}$ of the circle. Find the area and perimeter of the sector.

Solution: Since the sector is  $\frac{1}{4}$ of the circle, its area will be  $\frac{1}{4}$ the area of the circle. The area of the sector will be $\frac{\pi r^2}{4}$ .

$Area_{Sector}=\frac{\pi(4^2)}{4}=4 \pi \ in^2$

The perimeter of the sector is the sum of the lengths of the two radiuses and the arc. Each radius is 4 in . The arc is  $\frac{1}{4}$ of the circumference of the full circle. The length of the arc is $\frac{\pi d}{4}=\frac{\pi(8)}{4}=2 \pi$ .

$Perimeter_{Sector}=4+4+2 \pi=8+2 \pi \ in$

Concept Problem Revisited

You can understand the formula for the area of a circle by dissecting a circle into wedges and rearranging them to form a shape that is close to a parallelogram. The parallelogram can then by formed into a shape close to a rectangle.

The lengths of the sides of the “parallelogram” are  $r$ and $\frac{2 \pi r}{2}=\pi r$ . If you imagine cutting the wedges smaller and smaller, the parallelogram will look closer and closer to a rectangle with dimensions  $\pi r$ and $r$ . The area of the rectangle and thus the area of the circle will be $(\pi r)(r)=\pi r^2$ .

#### Vocabulary

circle is the set of all points equidistant from a center point.

Equidistant means the same distance away from.

Circumference is the perimeter of a circle.

$\pi$  is a Greek letter that stands for an irrational number that is approximately equal to 3.14.

The  radius of a circle is the distance from the center point to the circle.

The diameter of a circle is the distance from one side of the circle to the other while passing through the center point.

#### Guided Practice

1. Explain why the formula  $C=2 \pi r$ also works for the circumference of a circle.

2. Find the area of a circle with a circumference of $12 \pi \ cm$ .

3. Find the area and circumference of the sector below. The sector is  $\frac{1}{3}$ of the circle.

1. The relationship between the radius and the diameter is $d=2r$ . If you substitute  $2r$ for  $d$ in the formula $C=\pi d$ , you will have  $C=\pi 2r$ or $C=2 \pi r$ .

2. If the circumference is  $12 \pi \ cm$ then the diameter is  $12 \ cm$ and the radius is therefore $6 \ cm$ . The area is $A=\pi(6^2)=36 \pi \ cm^2$ .

3. $Area=\frac{\pi r^2}{3}=\frac{\pi(9^2)}{3}=27 \pi \ cm^2$ .

$Perimeter=r+r+\frac{\pi d}{3}=9+9+\frac{18 \pi}{3}=18+6 \pi \ cm$

#### Practice

1. Find the area and circumference of a circle with radius 3 in .

2. Find the area and circumference of a circle with radius 6 in .

3. Find the area and circumference of a circle with diameter 15 cm .

4. Find the area and circumference of a circle with diameter 22 in .

5. Find the area of a circle with a circumference of $32 \pi \ cm$ .

6. Find the circumference of a circle with area $32 \pi \ cm^2$ .

7. The sector below is  $\frac{1}{2}$ of a circle. Find the area of the sector.

8. Find the perimeter of the sector in #7.

9. The sector below is  $\frac{3}{4}$ of a circle. Find the area of the sector.

10. Find the perimeter of the sector in #9.

11. The sector below is  $\frac{1}{8}$ of a circle. Find the area of the sector.

12. Find the perimeter of the sector in #11.

13. The sector below is  $\frac{2}{3}$ of a circle. Find the area of the sector.

14. Find the perimeter of the sector in #13.

15. Explain the formula  $A=\frac{Cr}{2}$ where  $A$ is the area of a circle,  $C$ is the circumference of the circle, and  $r$ is the radius of the circle.