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# 8.5: Sector Area

Difficulty Level: 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:

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

• The sector is m360\begin{align*}\frac{m}{360}\end{align*} of the circle.
• The sector is also sC=s2πr\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 πr2\begin{align*}\pi r^2\end{align*}. Let A\begin{align*}A\end{align*} be the area of the sector:

A=m360πr2\begin{align*}A = \frac{m}{360} \cdot \pi r^2\end{align*}

Also,

A=sCπr2=s2πrπr2=12sr\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 r\begin{align*}r\end{align*}. A sector of the circle has an arc with degree measure m\begin{align*}m^\circ\end{align*} and arc length s\begin{align*}s\end{align*} units. The area of the sector is A\begin{align*}A\end{align*} square units:

A=m360πr2=12sr\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 90\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 :

36090=270\begin{align*}360^\circ - 90^\circ = 270^\circ\end{align*}

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

The area of the sector

A=m360πr2=270360π(6)2=3436π=27π square inches84.8 in2\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*}

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 360\begin{align*}360^\circ\end{align*}.

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