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?

#### Watch This

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

https://www.youtube.com/watch?v=zD4CsKIYEHo James Sousa: Arc Length and Area of a Sector

#### 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 @$\begin{align*}r\end{align*}@$ 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 examples, 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.

**Example A**

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.

**Solution:** 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.

**Example B**

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.

**Solution:** 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:

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

**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 @$\begin{align*}1 \ degree=\left(\frac{\pi}{180} \right) radians\end{align*}@$. This means that @$\begin{align*}64^\circ=64 \cdot \left(\frac{\pi}{180} \right) \approx 1.12 \ radians\end{align*}@$.

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

**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 @$\begin{align*}15^\circ\end{align*}@$. 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 @$\begin{align*}15^\circ=15 \cdot \left(\frac{\pi}{180} \right) \approx 0.26 \ radians\end{align*}@$.

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

#### 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):

**Answers:**

1. 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 @$\begin{align*}100^\circ=100 \cdot \left(\frac{\pi}{180} \right) \approx 1.75 \ radians\end{align*}@$.

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

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.

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

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.

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

#### Practice

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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