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# Area of Sectors and Segments

## Area of parts of a circle.

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Sector Area

You make a 10″ diameter cheesecake for a party and coat the top in a layer of chocolate. You divide the cheesecake up into equal slices so that the central angle of each slice is 15\begin{align*}15^\circ\end{align*}. How many square inches of chocolate are on the top of each slice?

#### Watch This

Watch the first part of this video starting at 5:40:

#### Guidance

A sector is a portion of a filled circle bounded by two radii and an arc. A sector is like a “wedge” of a circle. Below, the portion of the circle shaded red is a sector.

Because a sector is two dimensional, you can calculate its area. The area of a whole circle with radius r\begin{align*}r\end{align*} is πr2\begin{align*}\pi r^2\end{align*}. The area of a sector represents a fraction of this whole circle area. The measure of the central angle helps to tell you what fraction of the circle the sector is. In the examples, you will show that Sector Area=r2θ2\begin{align*}Sector \ Area =\frac{r^2 \theta}{2}\end{align*}, where r\begin{align*}r\end{align*} is the radius of the circle and θ\begin{align*}\theta\end{align*} is the measure of the central angle in radians.

Example A

In terms of θ\begin{align*}\theta\end{align*}, what fraction of the circle is the red sector? Assume θ\begin{align*}\theta\end{align*} is in radians.

Solution: Thinking in radians, a full circle is 2π\begin{align*}2 \pi\end{align*} radians. The sector is therefore θ2π\begin{align*}\frac{\theta}{2 \pi}\end{align*} of the whole circle.

Example B

Explain why the area of the sector below is r2θ2\begin{align*}\frac{r^2 \theta}{2}\end{align*} where θ\begin{align*}\theta\end{align*} is the measure of the central angle in radians.

Solution: The area of the whole circle is πr2\begin{align*}\pi r^2\end{align*} and this sector represents θ2π\begin{align*}\frac{\theta}{2 \pi}\end{align*} of the whole circle. Therefore, the area of the sector is:

πr2θ2π=r2θ2

Example C

Find the area of the red sector below.

Solution: The angle is given in degrees, so first convert it to radians. Remember that 1 degree=(π180)radians\begin{align*}1 \ degree=\left(\frac{\pi}{180} \right) radians\end{align*}. This means that 64=64(π180)1.12 radians\begin{align*}64^\circ=64 \cdot \left(\frac{\pi}{180} \right) \approx 1.12 \ radians\end{align*}.

Sector Area=r2θ2(32)(1.12)25.03 in2

Concept Problem Revisited

You make a 10″ diameter cheesecake for a party and coat the top in a layer of chocolate. You divide the cheesecake up into equal slices so that the central angle of each slice is 15\begin{align*}15^\circ\end{align*}. How many square inches of chocolate are on each slice?

The top of each slice covered in chocolate is a sector with radius 5 inches and a central angle of 15\begin{align*}15^\circ\end{align*}. To find the area of the sector, first convert 15\begin{align*}15^\circ\end{align*} to radians. Remember that 1 degree=(π180)radians\begin{align*}1 \ degree=\left(\frac{\pi}{180} \right) radians\end{align*}. This means that 15=15(π180)0.26 radians\begin{align*}15^\circ=15 \cdot \left(\frac{\pi}{180} \right) \approx 0.26 \ radians\end{align*}.

Chocolate Area=r2θ2(52)(0.26)23.27 in2

#### Vocabulary

A radian is the measure of the central angle that creates an arc with a length equal to the radius.

Arc length is the measure of the distance along the circle between the two endpoints that define an arc.

A sector is a portion of a filled circle bounded by two radii and an arc. A sector is like a “wedge” of a circle.

#### Guided Practice

1. Find the area of the sector below:

2. Find the area of the triangle below:

3. Use your answers to #1 and #2 to find the area of the circular segment below (shaded in purple):

1. To find the area of the sector, first convert 100\begin{align*}100^\circ\end{align*} to radians. Remember that 1 degree=(π180) radians\begin{align*}1 \ degree=\left( \frac{\pi}{180} \right) \ radians\end{align*}. This means that 100=100(π180)1.75 radians\begin{align*}100^\circ=100 \cdot \left(\frac{\pi}{180} \right) \approx 1.75 \ radians\end{align*}.

Sector Area=r2θ2(7)2(1.75)242.76 cm2

2. Because the radius of the circle is 7 cm, two sides of the triangle are length 7 cm.

You know two sides and an included angle so you can find the area using the sine area formula.

Triangle AreaTriangle AreaTriangle Area=12(a)(b)(sinC)=12(7)(7)sin10024.12 cm2

3. The circular segment is the portion of the sector not included in the triangle. To find its area, subtract the area of the triangle from the area of the sector.

Segment Area=42.76 cm224.12 cm2=18.64 cm2

#### Practice

1. Explain why the area of a sector is r2θ2\begin{align*}\frac{r^2 \theta}{2}\end{align*} where θ\begin{align*}\theta\end{align*} is the measure of the central angle in radians.

2. How do you find the area of a sector if the central angle is given in degrees?

Find the area of each region shaded in blue.

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### Vocabulary Language: English

Arc

Arc

An arc is a section of the circumference of a circle.
arc length

arc length

In calculus, arc length is the length of a plane function curve over an interval.

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.
Scale Factor

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.
Sector

Sector

A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees.
Sector of a Circle

Sector of a Circle

A sector of a circle is the area bounded by two radii and the arc between the endpoints of the radii.