# 8.5: Sector Area

**At Grade**Created by: CK-12

## Learning Objectives

- Calculate the area of a
*sector.*

## Area of a Sector

A **sector** is a section of a circle. Think of it like a pie slice: it is a section bounded on two sides by radii and one side by an arc. The tip of a sector is at the center of the circle:

image from http://en.wikibooks.org/wiki/Geometry/Circles/Sectors

- A
**sector**is just like a slice of ________________ or pizza.

The *area* of a **sector** is simply an appropriate fractional part of the area of the circle.

Suppose a **sector** of a circle with radius \begin{align*}r\end{align*} and circumference \begin{align*}C\end{align*} has an arc with a degree measure of \begin{align*}m^\circ\end{align*} and an arc length of \begin{align*}s\end{align*} units.

- The sector is \begin{align*}\frac{m}{360}\end{align*} of the circle.
- The sector is also \begin{align*}\frac{s}{C} = \frac{s}{2 \pi r}\end{align*} of the circle.

To find the *area* of the **sector,** just find one of these fractional parts of the *area* of the *circle.* We know that the *area* of the *circle* is \begin{align*}\pi r^2\end{align*}. Let \begin{align*}A\end{align*} be the *area* of the **sector:**

\begin{align*}A = \frac{m}{360} \cdot \pi r^2\end{align*}

Also,

\begin{align*}A = \frac{s}{C} \cdot \pi r^2 = \frac{s}{2 \pi r} \cdot \pi r^2 = \frac{1}{2}sr\end{align*}

**Area of a Sector**

A circle has radius \begin{align*}r\end{align*}. A sector of the circle has an arc with degree measure \begin{align*}m^\circ\end{align*} and arc length \begin{align*}s\end{align*} units. The area of the sector is \begin{align*}A\end{align*} square units:

\begin{align*}A = \frac{m}{360} \cdot \pi r^2 = \frac{1}{2} sr\end{align*}

This means that the *area* of a **sector** is one-half of the product of the ________________ and the arc length.

**Example 1**

*Mark drew a sheet metal pattern made up of a circle with a sector cut out. The pattern is made from an arc of a circle and two perpendicular 6 inch radii like so:*

*How much sheet metal does Mark need for the pattern?*

The cut-out **sector** has a **degree measure** of \begin{align*}90^\circ\end{align*} because the radii are *perpendicular*.

The measure of the arc of the metal piece is (the entire circle) – (the cut-out sector) or :

\begin{align*}360^\circ - 90^\circ = 270^\circ\end{align*}

Using the values radius \begin{align*}r =\end{align*} _________ and arc degree measure \begin{align*}m = \underline{\;\;\;\;\;\;\;\;\;\;\;}^\circ\end{align*},

The *area* of the sector

\begin{align*}A& = \frac{m}{360} \cdot \pi r^2\\ &= \frac{270}{360} \cdot \pi (6)^2 = \frac{3}{4} \cdot 36 \pi = 27 \pi \ \text{square inches}\\ &\approx 84.8 \ in^2\end{align*}

**Reading Check:**

*Explain the following statement in your own words:*

The area of a sector is a fraction of the area of a circle, and the fraction is calculated by the degree measure of the sector divided by \begin{align*}360^\circ\end{align*}.

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