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Parts of Circles

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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.

Answers:

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.

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