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# 2.6: Area of a Sector

Difficulty Level: At Grade Created by: CK-12
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While eating lunch with your friends one day you decide to get some pie. The school cafeteria has a sale on pies if you buy the whole pie instead of an individual slice. Since you and your friends plan to eat the whole pie anyway, you are happy to make the purchase.

You decide to cut yourself a piece. When you have taken your section of pie out, one of your friends objects that you have cut out more than your fair share. If the radius of the pie is six inches, and the angle formed by the end of your pie wedge is 30\begin{align*}30^\circ\end{align*}, what is the area of your pie piece? Is it more than your fair share if there are five people (including you) splitting the pie?

### Watch This

In the second part of this video, you'll see how to find the area of a sector in a circle.

### Guidance

One of the most common geometric formulas is the area of a circle:

A=πr2\begin{align*}A = \pi r^2\end{align*}

In terms of angle rotation, this is the area created by 2π\begin{align*}2 \pi\end{align*} radians.

2π rad=πr2 area\begin{align*}2\pi \ \text{rad} = \pi r^2 \ \text{area}\end{align*}

A half-circle, or π\begin{align*}\pi\end{align*} radian rotation would create a section, or sector of the circle equal to half the area or:

12πr2\begin{align*}\frac{1}{2}\pi r^2\end{align*}

So an angle of 1 radian would define an area of a sector equal to:

1=12r2\begin{align*}1=\frac{1}{2} r^2\end{align*}

From this we can determine the area of the sector created by any angle, θ\begin{align*}\theta\end{align*} radians, to be:

A=12r2θ\begin{align*}A=\frac{1}{2}r^2 \theta\end{align*}

#### Example A

Crops are often grown using a technique called center pivot irrigation that results in circular shaped fields.

Here is a satellite image taken over fields in Kansas that use this type of irrigation system.

If the irrigation pipe is 450 m in length, what is the area that can be irrigated after a rotation of 2π3\begin{align*}\frac{2\pi}{3}\end{align*} radians?

Solution: Using the formula:

AA=12r2θ=12(450)2(2π3)\begin{align*}A&=\frac{1}{2}r^2 \theta\\ A&=\frac{1}{2}(450)^2 \left(\frac{2\pi}{3}\right)\end{align*}

The area is approximately 212,058 square meters.

#### Example B

A doughnut has a hole in the middle with a radius of 1 cm, and the distance from the center of the hole to the outer edge of the doughnut is 3 cm. What is the area of a sector of 14\begin{align*}\frac{1}{4}\end{align*} of the doughnut?

Solution: The formula for the area of a sector is A=12r2θ\begin{align*}A&=\frac{1}{2}r^2 \theta\end{align*}. Using this formula to find the area of the sector from the center outward gives:

AAA=12r2θ=1232π2=9π4\begin{align*} A&=\frac{1}{2}r^2 \theta\\ A&=\frac{1}{2}3^2 \frac{\pi}{2}\\ A&=\frac{9 \pi}{4} \end{align*}

Now it is necessary to subtract the area of the sector that is part of the hole, and therefore not part of the doughnut:

AAA=12r2θ=12(1)2π2=π4\begin{align*} A&=\frac{1}{2}r^2 \theta\\ A&=\frac{1}{2}(1)^2 \frac{\pi}{2}\\ A&=\frac{\pi}{4} \end{align*}

Area of the sector of doughnut: A=9π4π4=8π4=2π\begin{align*} A&=\frac{9 \pi}{4}-\frac{\pi}{4} = \frac{8 \pi}{4} = 2\pi \end{align*}

#### Example C

A driver is traveling around a circular track that has radius of 70 meters. If the angle from the starting line to her current position is π3\begin{align*}\frac{\pi}{3}\end{align*} radians, what is the area of the sector traced out by her car?

Solution: The area of a sector is: A=12r2θ\begin{align*}A&=\frac{1}{2}r^2 \theta\end{align*}. This leads us to:

AAA=12r2θ=12(70)2π373.3\begin{align*} A&=\frac{1}{2}r^2 \theta\\ A&=\frac{1}{2}(70)^2 \frac{\pi}{3}\\ A & \approx 73.3\\ \end{align*} meters

### Vocabulary

Sector: A sector is the portion of a circle between two lines from the origin to the circle's edge.

### Guided Practice

1. If the radius of a sector is 5 feet, and the sector sweeps out an angle of 43\begin{align*}43^\circ\end{align*}, find the area of the sector.

2. If a pie wedge has an area of 15 square in inches, and the pie has a radius of 9 inches, find the angle swept out by the sector.

3. If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of 25\begin{align*}25^\circ\end{align*}, find the radius of the cake.

Solutions:

1. Since you know that A=12r2θ\begin{align*}A = \frac{1}{2} r^2 \theta\end{align*}, you can solve for the area (don't forget to convert the degrees of the angle to radians):

A=12r2θA=12(25)(.75)A=9.375\begin{align*} A = \frac{1}{2} r^2 \theta\\ A = \frac{1}{2} (25) (.75)\\ A = 9.375\\ \end{align*}

2. Since you know that A=12r2θ\begin{align*}A = \frac{1}{2} r^2 \theta\end{align*}, you can solve for the angle swept out by the sector (don't forget that the angle will be measured in radians):

A=12r2θ15=12(81)(θ)θ=(2)(15)81θ=.37\begin{align*} A = \frac{1}{2} r^2 \theta\\ 15 = \frac{1}{2} (81) (\theta)\\ \theta = \frac{(2)(15)}{81}\\ \theta = .37\\ \end{align*}

3. Since you know that A=12r2θ\begin{align*}A = \frac{1}{2} r^2 \theta\end{align*}, you can solve for the radius of the cake (don't forget to convert the degrees of the angle to radians):

A=12r2θ20=12(r2)(.436)r2=(2)(20).436r2=91.743r=9.58\begin{align*} A = \frac{1}{2} r^2 \theta\\ 20 = \frac{1}{2} (r^2) (.436)\\ r^2 = \frac{(2)(20)}{.436}\\ r^2 = 91.743\\ r = 9.58\\ \end{align*}

### Concept Problem Solution

You now know that the equation for the area of a circle swept out by some angle is:

A=12r2θ\begin{align*}A=\frac{1}{2}r^2 \theta\end{align*}

Applying this to your pie slice, you have an area of:

A=12×62×π6=3π\begin{align*}A=\frac{1}{2} \times 6^2 \times \frac{\pi}{6} = 3\pi\end{align*}

And the total area of the pie is:

πr2=36π\begin{align*}\pi r^2 = 36\pi\end{align*}

To find out what your fair portion of the pie is, multiply the total area by your fraction:

15×36π=7.2π\begin{align*}\frac{1}{5} \times 36\pi = 7.2\pi\end{align*}

And since the piece you took is only 3π\begin{align*}3\pi\end{align*} in size, you are definitely not taking too much!

### Practice

1. If the radius of a sector is 8 inches, and the central angle of the sector is 40\begin{align*}40^\circ\end{align*}, find the area of the sector.
2. If the radius of a sector is 12 inches, and the central angle of the sector is π6\begin{align*}\frac{\pi}{6}\end{align*} radians, find the area of the sector.
3. If the radius of a sector is 6 inches, and the central angle of the sector is 140\begin{align*}140^\circ\end{align*}, find the area of the sector.
4. If the radius of a sector is 5 inches, and the central angle of the sector is 5π3\begin{align*}\frac{5\pi}{3}\end{align*} radians, find the area of the sector.
5. If the radius of a sector is 10 inches, and the central angle of the sector is 100\begin{align*}100^\circ\end{align*}, find the area of the sector.
6. If a pie wedge has an area of 10 square inches, and the pie has a radius of 6 inches, find the angle swept out by the sector.
7. If a pie wedge has an area of 15 square inches, and the pie has a radius of 4 inches, find the angle swept out by the sector.
8. If a pie wedge has an area of 12 square inches, and the pie has a radius of 3 inches, find the angle swept out by the sector.
9. If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of π3\begin{align*}\frac{\pi}{3}\end{align*} radians, find the radius of the cake.
10. If you have a piece of round cake that has an area of 100 square inches, and you know the piece sweeps out an angle of 50\begin{align*}50^\circ\end{align*}, find the radius of the cake.
11. If you have a piece of round cake that has an area of 35 square inches, and you know the piece sweeps out an angle of 2π5\begin{align*}\frac{2\pi}{5}\end{align*} radians, find the radius of the cake.
12. If you have a piece of round cake that has an area of 20 square inches, and you know the piece sweeps out an angle of 30\begin{align*}30^\circ\end{align*}, find the radius of the cake.

A pizza has a radius of 10in. Use this information to answer questions 13-15.

1. A slice is removed. The length of the crust of the missing slice is 3in. What is the area of the missing slice?
2. You eat three pieces with a central angle of 4π5\begin{align*}\frac{4\pi}{5}\end{align*}. What is the area of the pizza you ate?
3. A large pizza has a radius of 12in. What is the area of half of the large pizza?

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### Vocabulary Language: English

TermDefinition
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.

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