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Intermediate Geometry : How to find the area of a sector

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Example Questions

Example Question #1 : How To Find The Area Of A Sector

Find the approximate area of the shaded portion of the figure.

Circle1

Possible Answers:

$6200\ cm^{2}$ $\displaystyle 6200\ cm^{2}$

$4200\ cm^{2}$ $\displaystyle 4200\ cm^{2}$

$3200\ cm^{2}$ $\displaystyle 3200\ cm^{2}$

$3900\ cm^{2}$ $\displaystyle 3900\ cm^{2}$

$7100\ cm^{2}$ $\displaystyle 7100\ cm^{2}$

Correct answer:

$6200\ cm^{2}$ $\displaystyle 6200\ cm^{2}$

Explanation:

The answer is approximately $6200\ cm^{2}$ $\displaystyle 6200\ cm^{2}$.

First, you would need to find the diameter of the circle. Use the Pythagorean Theorem to get

$50^{2}+120^{2}=130^{2}$ $\displaystyle 50^{2}+120^{2}=130^{2}$

$78^{2}+104^{2}=130^{2}$ $\displaystyle 78^{2}+104^{2}=130^{2}$

Since the diameter is 130, we divide by 2 to get 65 cm for our radius. Then the area of the circle is

$\pi r^{2} = \pi \cdot 65^{2} \approx 13,273 \ cm^{2}$ $\displaystyle \pi r^{2} = \pi \cdot 65^{2} \approx 13,273 \ cm^{2}$

Next we would find the area of each triangle:

$\frac{1}{2} (50)(120) = 3000cm^{2}$ $\displaystyle \frac{1}{2} (50)(120) = 3000cm^{2}$

and

$\frac{1}{2} (78)(104) = 4056cm^{2}$ $\displaystyle \frac{1}{2} (78)(104) = 4056cm^{2}$

Then we would subtract these from our answer above to get:

$13273-3000 - 4056 \approx 6200\ cm^{2}$ $\displaystyle 13273-3000 - 4056 \approx 6200\ cm^{2}$.

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Example Question #2 : How To Find The Area Of A Sector

Circle

The radius of the circle above is $\displaystyle 4$ and $\angle A=45^{\circ}$ $\displaystyle \angle A=45^{\circ}$. What is the area of the shaded section of the circle?

Possible Answers:

$\pi$ $\displaystyle \pi$

$2\pi$ $\displaystyle 2\pi$

$8\pi$ $\displaystyle 8\pi$

$16\pi$ $\displaystyle 16\pi$

$4\pi$ $\displaystyle 4\pi$

Correct answer:

$2\pi$ $\displaystyle 2\pi$

Explanation:

Area of Circle = πr² = π4²= 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

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Example Question #1 : How To Find The Area Of A Sector

A circle has a diameter of $\displaystyle 60$ meters. A certain sector of the circle has a central angle of $\small 20^\circ$ $\displaystyle \small 20^\circ$. Find the area of the sector.

Possible Answers:

$\small 200\pi\,m^2$ $\displaystyle \small 200\pi\,m^2$

$\small 50\pi\,m^2$ $\displaystyle \small 50\pi\,m^2$

$\small 25\pi\,m^2$ $\displaystyle \small 25\pi\,m^2$

$\small 30\pi\,m^2$ $\displaystyle \small 30\pi\,m^2$

$\small 6\pi\,m^2$ $\displaystyle \small 6\pi\,m^2$

Correct answer:

$\small 50\pi\,m^2$ $\displaystyle \small 50\pi\,m^2$

Explanation:

The formula for the area of a sector is.

$\small S=\frac{m}{360}\pi r^2$ $\displaystyle \small S=\frac{m}{360}\pi r^2$ where $\small r$ $\displaystyle \small r$ is the radius and $\small m$ $\displaystyle \small m$ is the measure of the central angle of the sector.

We are given that the diameter of the circle is 60. Therefore its radius is simply half as long, or 30.

Substituting into our equation gives

$\small \small S=\frac{20}{360}\pi(30^2)=\frac{1}{18}\pi(900)=50\pi$ $\displaystyle \small \small S=\frac{20}{360}\pi(30^2)=\frac{1}{18}\pi(900)=50\pi$

Therefore our area is $\small 50\pi\,m^2$ $\displaystyle \small 50\pi\,m^2$

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Example Question #32 : Circles

Find the area of a sector with a central angle of 124 $\displaystyle 124$ degrees and a radius of $\displaystyle 12$.

Possible Answers:

155.82 $\displaystyle 155.82$

148.20 $\displaystyle 148.20$

159.22 $\displaystyle 159.22$

156.29 $\displaystyle 156.29$

Correct answer:

155.82 $\displaystyle 155.82$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{124}{360}\times\pi\times 12^2$ $\displaystyle \text{Area of Sector}=\frac{124}{360}\times\pi\times 12^2$

Solve and round to two decimal places.

$\text{Area of Sector}=155.82$ $\displaystyle \text{Area of Sector}=155.82$

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Example Question #3 : How To Find The Area Of A Sector

Find the area of a sector that has a central angle of 129 $\displaystyle 129$ degrees and a radius of $\displaystyle 13$.

Possible Answers:

190.25 $\displaystyle 190.25$

192.12 $\displaystyle 192.12$

182.46 $\displaystyle 182.46$

200.01 $\displaystyle 200.01$

Correct answer:

190.25 $\displaystyle 190.25$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{129}{360}\times\pi\times 13^2$ $\displaystyle \text{Area of Sector}=\frac{129}{360}\times\pi\times 13^2$

Solve and round to two decimal places.

$\text{Area of Sector}=190.25$ $\displaystyle \text{Area of Sector}=190.25$

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Example Question #1 : How To Find The Area Of A Sector

Find the area of a sector that has a central angle of 130 $\displaystyle 130$ degrees and a radius of $\displaystyle 19$.

Possible Answers:

409.54 $\displaystyle 409.54$

485.12 $\displaystyle 485.12$

412.29 $\displaystyle 412.29$

399.50 $\displaystyle 399.50$

Correct answer:

409.54 $\displaystyle 409.54$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{130}{360}\times\pi\times 19^2$ $\displaystyle \text{Area of Sector}=\frac{130}{360}\times\pi\times 19^2$

Solve and round to two decimal places.

$\text{Area of Sector}=409.54$ $\displaystyle \text{Area of Sector}=409.54$

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Example Question #1 : How To Find The Area Of A Sector

Find the area of a sector that has a central angle of 135 $\displaystyle 135$ degrees and a radius of $\displaystyle 21$.

Possible Answers:

529.30 $\displaystyle 529.30$

519.54 $\displaystyle 519.54$

527.02 $\displaystyle 527.02$

609.39 $\displaystyle 609.39$

Correct answer:

519.54 $\displaystyle 519.54$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{135}{360}\times\pi\times 21^2$ $\displaystyle \text{Area of Sector}=\frac{135}{360}\times\pi\times 21^2$

Solve and round to two decimal places.

$\text{Area of Sector}=519.54$ $\displaystyle \text{Area of Sector}=519.54$

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Example Question #1 : How To Find The Area Of A Sector

Find the area of a sector that has a central angle of 115 $\displaystyle 115$ degrees and a radius of $\displaystyle 8$.

Possible Answers:

60.39 $\displaystyle 60.39$

70.05 $\displaystyle 70.05$

64.23 $\displaystyle 64.23$

69.21 $\displaystyle 69.21$

Correct answer:

64.23 $\displaystyle 64.23$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{115}{360}\times\pi\times 8^2$ $\displaystyle \text{Area of Sector}=\frac{115}{360}\times\pi\times 8^2$

Solve and round to two decimal places.

$\text{Area of Sector}=64.23$ $\displaystyle \text{Area of Sector}=64.23$

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Example Question #32 : Plane Geometry

Find the area of a sector that has a central angle of 118 $\displaystyle 118$ degrees and a radius of $\displaystyle 10$.

Possible Answers:

98.45 $\displaystyle 98.45$

101.59 $\displaystyle 101.59$

99.21 $\displaystyle 99.21$

102.97 $\displaystyle 102.97$

Correct answer:

102.97 $\displaystyle 102.97$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{118}{360}\times\pi\times 10^2$ $\displaystyle \text{Area of Sector}=\frac{118}{360}\times\pi\times 10^2$

Solve and round to two decimal places.

$\text{Area of Sector}=102.97$ $\displaystyle \text{Area of Sector}=102.97$

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Example Question #1 : How To Find The Area Of A Sector

Find the area of a sector that has a central angle of $\displaystyle 34$ degrees and a radius of $\displaystyle 5$.

Possible Answers:

12.66 $\displaystyle 12.66$

6.10 $\displaystyle 6.10$

16.92 $\displaystyle 16.92$

7.42 $\displaystyle 7.42$

Correct answer:

7.42 $\displaystyle 7.42$

Explanation:

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{34}{360}\times\pi\times 5^2$ $\displaystyle \text{Area of Sector}=\frac{34}{360}\times\pi\times 5^2$

Solve and round to two decimal places.

$\text{Area of Sector}=7.42$ $\displaystyle \text{Area of Sector}=7.42$

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Intermediate Geometry : How to find the area of a sector

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Area Of A Sector

Example Question #2 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #32 : Circles

Example Question #3 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #32 : Plane Geometry

Example Question #1 : How To Find The Area Of A Sector

All Intermediate Geometry Resources

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