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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #61 : Circles

Find the area of a sector if it has an arc length of $3\pi$ and a radius of 5.7 .

Possible Answers:

26.86

23.11

25.99

24.09

Correct answer:

26.86

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{3\pi(5.7)}{2}=26.86$

Make sure to round to places after the decimal.

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Example Question #61 : Intermediate Geometry

Find the area of a sector if it has an arc length of $10\pi$ and a radius of .

Possible Answers:

171.41

192.59

186.21

188.50

Correct answer:

188.50

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{12\pi(10)}{2}=188.50$

Make sure to round to places after the decimal.

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Example Question #32 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of $15\pi$ and a radius of 10.4 .

Possible Answers:

245.04

251.59

214.81

199.44

Correct answer:

245.04

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{15\pi(10.4)}{2}=245.04$

Make sure to round to places after the decimal.

Report an Error

Example Question #31 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of $19\pi$ and a radius of .

Possible Answers:

1521.20

1428.58

1492.26

1304.51

Correct answer:

1492.26

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{19\pi(50)}{2}=1492.26$

Make sure to round to places after the decimal.

Report an Error

Example Question #33 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of and a radius of $12\pi$ .

Possible Answers:

77.91

81.08

75.40

90.36

Correct answer:

75.40

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{4(12\pi)}{2}=75.40$

Make sure to round to places after the decimal.

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Example Question #66 : Circles

Find the area of a sector if it has an arc length of and a radius of $20\sqrt3$ .

Possible Answers:

233.90

259.81

241.20

215.29

Correct answer:

259.81

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{15(20\sqrt3)}{2}=259.81$

Make sure to round to places after the decimal.

Report an Error

Example Question #67 : Circles

Find the area of a sector if it has an arc length of $13\pi$ and a radius of $49\sqrt3$ .

Possible Answers:

1441.20

1733.09

1720.91

1695.48

Correct answer:

1733.09

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{13\pi(49\sqrt3)}{2}=1733.09$

Make sure to round to places after the decimal.

Report an Error

Example Question #32 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of and a radius of $20\sqrt2$ .

Possible Answers:

268.70

255.64

210.40

249.17

Correct answer:

268.70

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{19(20\sqrt2)}{2}=268.70$

Make sure to round to places after the decimal.

Report an Error

Example Question #33 : How To Find The Area Of A Sector

Find the area of a sector if it has a central angle of 172 degrees and a radius of .

Possible Answers:

384.25

891.41

701.02

722.39

Correct answer:

384.25

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ , where is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{172}{360}\pi(16)^2=384.25$

Make sure to round to two places after the decimal.

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Example Question #41 : How To Find The Area Of A Sector

Give the area of a sector of a circle if its central angle is $20^{\circ }$ and the length of its arc is 10 inches. Write the answer in terms of $\pi$ , if applicable.

Possible Answers:

$\frac{450}{\pi }\;in^{2}$

$\frac{\sqrt{450}}{\pi }\;in^{2}$

$300\pi \; in^{2}$

$\frac{900}{\pi }\;in^{2}$

$450\pi \; in^{2}$

Correct answer:

$\frac{450}{\pi }\;in^{2}$

Explanation:

The arc of the $20^{\circ }$ sector measures 10 inches, so the circumference of the entire circle is

$C=10 \cdot\frac{360}{20}=180$

The radius is therefore

$r=\frac{C}{2\pi }=\frac{180}{2\pi }=\frac{90}{\pi }$

The area of a $20^{\circ }$ sector of this circle measures:

$A = \frac{20}{360}\cdot\pi r^{2}= \frac{20}{360}\cdot\pi \cdot \left (\frac{90}{\pi } \right )^{2}$

$=\frac{20\cdot 90\cdot 90\cdot \pi }{360\cdot \pi \cdot \pi }=\frac{450}{\pi }$

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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #61 : Circles

Example Question #61 : Intermediate Geometry

Example Question #32 : How To Find The Area Of A Sector

Example Question #31 : How To Find The Area Of A Sector

Example Question #33 : How To Find The Area Of A Sector

Example Question #66 : Circles

Example Question #67 : Circles

Example Question #32 : How To Find The Area Of A Sector

Example Question #33 : How To Find The Area Of A Sector

Example Question #41 : How To Find The Area Of A Sector

All Intermediate Geometry Resources

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