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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 \begin{align*}15^\circ\end{align*}. How many square inches of chocolate are on the top of each slice?

Sector Area

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 \begin{align*}r\end{align*} is \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 problems below, you will show that \begin{align*}Sector \ Area =\frac{r^2 \theta}{2}\end{align*}, where \begin{align*}r\end{align*} is the radius of the circle and \begin{align*}\theta\end{align*} is the measure of the central angle in radians.

 

Let's take at some problems about sector area.  

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

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

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

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

\begin{align*}\pi r^2 \cdot \frac{\theta}{2 \pi}=\frac{r^2 \theta}{2}\end{align*}

3. Find the area of the red sector below.

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

\begin{align*}Sector \ Area=\frac{r^2 \theta}{2} \approx \frac{(3^2)(1.12)}{2} \approx 5.03 \ in^2\end{align*}

Examples

Example 1

Earlier, you were given a problem about dividing a cheesecake. 

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 \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 \begin{align*}15^\circ\end{align*}. To find the area of the sector, first convert \begin{align*}15^\circ\end{align*} to radians. Remember that \begin{align*}1 \ degree=\left(\frac{\pi}{180} \right) radians\end{align*}. This means that \begin{align*}15^\circ=15 \cdot \left(\frac{\pi}{180} \right) \approx 0.26 \ radians\end{align*}.

\begin{align*}Chocolate \ Area=\frac{r^2 \theta}{2} \approx \frac{(5^2)(0.26)}{2} \approx 3.27 \ in^2\end{align*}

Example 2

Find the area of the sector below:

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

\begin{align*}Sector \ Area=\frac{r^2 \theta}{2} \approx \frac{(7)^2(1.75)}{2} \approx 42.76 \ cm^2\end{align*}

Example 3

Find the area of the triangle below:

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.

\begin{align*}Triangle \ Area &=\frac{1}{2}(a)(b)(\sin C) \\ Triangle \ Area &=\frac{1}{2}(7)(7)\sin 100^\circ \\ Triangle \ Area & \approx 24.12 \ cm^2\end{align*}

Example 4

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

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.

\begin{align*}Segment \ Area=42.76 \ cm^2 - 24.12 \ cm^2=18.64 \ cm^2\end{align*}

Review

1. Explain why the area of a sector is \begin{align*}\frac{r^2 \theta}{2}\end{align*} where \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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Review (Answers)

To see the Review answers, open this PDF file and look for section 8.10. 

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Vocabulary

Arc

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

arc length

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

radian

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

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

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

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

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