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# Parts of Circles

## Radius, chord, diameter, secant, and tangent.

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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)\begin{align*}(r)\end{align*}. From one side of the circle to the other through the center point is called the diameter (d)\begin{align*}(d)\end{align*}.

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

C=πd\begin{align*}C=\pi d\end{align*}

The area of a circle is also related to π\begin{align*}\pi\end{align*}.

A=πr2\begin{align*}A=\pi r^2\end{align*}

Example A

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

Solution:

Area=πr2=π(52)=25π in2\begin{align*}Area =\pi r^2=\pi(5^2)=25 \pi \ in^2\end{align*}

Circumference=πd=π(2.5)=10π in\begin{align*}Circumference =\pi d=\pi(2.5)=10 \pi \ in\end{align*}

Leaving your answer with a π\begin{align*}\pi\end{align*} symbol in it is called leaving your answer “in terms of π\begin{align*}\pi\end{align*}”. This is often preferable because it is the exact answer. As soon as you approximate a value for π\begin{align*}\pi\end{align*} your answer is not exact. Keep in mind that π\begin{align*}\pi\end{align*} 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=πr2=π(82)=64π cm2\begin{align*}Area =\pi r^2=\pi(8^2)=64 \pi \ cm^2\end{align*}

Circumference=πd=16π cm\begin{align*}Circumference =\pi d=16 \pi \ cm\end{align*}

Example C

The shape below is a portion of a circle called a sector. This sector is 14\begin{align*}\frac{1}{4}\end{align*} of the circle. Find the area and perimeter of the sector.

Solution: Since the sector is 14\begin{align*}\frac{1}{4}\end{align*} of the circle, its area will be 14\begin{align*}\frac{1}{4}\end{align*} the area of the circle. The area of the sector will be πr24\begin{align*}\frac{\pi r^2}{4}\end{align*}.

AreaSector=π(42)4=4π in2\begin{align*}Area_{Sector}=\frac{\pi(4^2)}{4}=4 \pi \ in^2\end{align*}

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 14\begin{align*}\frac{1}{4}\end{align*} of the circumference of the full circle. The length of the arc is πd4=π(8)4=2π\begin{align*}\frac{\pi d}{4}=\frac{\pi(8)}{4}=2 \pi\end{align*}.

PerimeterSector=4+4+2π=8+2π in\begin{align*}Perimeter_{Sector}=4+4+2 \pi=8+2 \pi \ in\end{align*}

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\begin{align*}r\end{align*} and 2πr2=πr\begin{align*}\frac{2 \pi r}{2}=\pi r\end{align*}. If you imagine cutting the wedges smaller and smaller, the parallelogram will look closer and closer to a rectangle with dimensions πr\begin{align*}\pi r\end{align*} and r\begin{align*}r\end{align*}. The area of the rectangle and thus the area of the circle will be (πr)(r)=πr2\begin{align*}(\pi r)(r)=\pi r^2\end{align*}.

#### 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.

π\begin{align*}\pi\end{align*} 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=2πr\begin{align*}C=2 \pi r\end{align*} also works for the circumference of a circle.

2. Find the area of a circle with a circumference of 12π cm\begin{align*}12 \pi \ cm\end{align*}.

3. Find the area and circumference of the sector below. The sector is 13\begin{align*}\frac{1}{3}\end{align*} of the circle.

1. The relationship between the radius and the diameter is d=2r\begin{align*}d=2r\end{align*}. If you substitute 2r\begin{align*}2r\end{align*} for d\begin{align*}d\end{align*} in the formula C=πd\begin{align*}C=\pi d\end{align*}, you will have C=π2r\begin{align*}C=\pi 2r\end{align*} or C=2πr\begin{align*}C=2 \pi r\end{align*}.

2. If the circumference is 12π cm\begin{align*}12 \pi \ cm\end{align*} then the diameter is 12 cm\begin{align*}12 \ cm\end{align*} and the radius is therefore 6 cm\begin{align*}6 \ cm\end{align*}. The area is A=π(62)=36π cm2\begin{align*}A=\pi(6^2)=36 \pi \ cm^2\end{align*}.

3. Area=πr23=π(92)3=27π cm2\begin{align*}Area=\frac{\pi r^2}{3}=\frac{\pi(9^2)}{3}=27 \pi \ cm^2\end{align*}.

Perimeter=r+r+πd3=9+9+18π3=18+6π cm\begin{align*}Perimeter=r+r+\frac{\pi d}{3}=9+9+\frac{18 \pi}{3}=18+6 \pi \ cm\end{align*}

#### 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 32π cm\begin{align*}32 \pi \ cm\end{align*}.

6. Find the circumference of a circle with area 32π cm2\begin{align*}32 \pi \ cm^2\end{align*}.

7. The sector below is 12\begin{align*}\frac{1}{2}\end{align*} of a circle. Find the area of the sector.

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

9. The sector below is 34\begin{align*}\frac{3}{4}\end{align*} of a circle. Find the area of the sector.

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

11. The sector below is \begin{align*}\frac{1}{8}\end{align*} of a circle. Find the area of the sector.

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

13. The sector below is \begin{align*}\frac{2}{3}\end{align*} of a circle. Find the area of the sector.

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

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

### Vocabulary Language: English

Circle

Circle

A circle is the set of all points at a specific distance from a given point in two dimensions.
Circumference

Circumference

The circumference of a circle is the measure of the distance around the outside edge of a circle.
Diameter

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.

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

Secant

The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
Tangent

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.
Tangent Circles

Tangent Circles

Tangent Circles are two or more circles that intersect at one point.