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Intermediate Geometry : Circles

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

What is the angle of a sector of area in^2 on a circle having a radius of $15\:in$ ?

Possible Answers:

$15.22^{\circ}$

$0.06^{\circ}$

$22.92^{\circ}$

$3.00^{\circ}$

$72.00^{\circ}$

Correct answer:

$22.92^{\circ}$

Explanation:

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total 360 degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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

What is the angle of a sector that has an arc length of 13.5 on a circle of diameter ?

Possible Answers:

$35.81^{\circ}$

$14.24^{\circ}$

$10.74^{\circ}$

$194.14^{\circ}$

$128.92^{\circ}$

Correct answer:

$128.92^{\circ}$

Explanation:

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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

What is the sector angle, in degrees, if the area of the sector is $4\pi$ with a given radius of ?

Possible Answers:

$45^{\circ}$

$360^{\circ}$

$0^{\circ}$

$90^{\circ}$

$60^{\circ}$

Correct answer:

$90^{\circ}$

Explanation:

Write the formula for the area of a circular sector.

$A= \frac{\Theta }{360}\pi r^2$

Substitute the given information and solve for theta:

$4\pi= \frac{\Theta }{360}\pi (4)^2$

$\frac{1}{4}= \frac{\Theta }{360}$

$\Theta= \frac{360}{4} = 90^{\circ}$

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

The length of the intercepted arc of a sector of a circle with radius $\small 12$ meters is $\small \small 4\pi$ meters. Find the measure of the central angle of the sector.

Possible Answers:

$\small 135^\circ$

$\small 15^\circ$

$\small 45^\circ$

$\small 60^\circ$

$\small 30^\circ$

Correct answer:

$\small 60^\circ$

Explanation:

The formula for the length of an arc is

$\small s=\frac{m}{360}2\pi r$

where $\small m$ is measure of the central angle and $\small r$ is the radius. Substituting what we know gives.

$\small \small 4\pi=\frac{m}{360}(2\pi)(12)$

$\small \small 4\pi=\frac{m\pi}{15}$

$\small \small 60\pi=m\pi$

$\small \small m=60$

Therefore, our central angle has a measure of $\small 60^\circ$

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

A sector in a circle with a radius of has an area of $12\pi$ . In degrees, what is the measurement of the central angle for this sector?

Possible Answers:

$45^\circ$

$15^\circ$

$60^\circ$

$30^\circ$

Correct answer:

$30^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$

The central angle is degrees.

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Example Question #4 : How To Find The Angle Of A Sector

A sector in a circle with a radius of has an area of $12\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$285^\circ$

$255^\circ$

$270^\circ$

$240^\circ$

Correct answer:

$270^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 4^2}=\frac{4320\pi}{16\pi}=270$

The central angle is 270 degrees.

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Example Question #5 : How To Find The Angle Of A Sector

A sector in a circle with a radius of has an area of $\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$5^\circ$

$10^\circ$

$15^\circ$

$20^\circ$

Correct answer:

$10^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times \pi}{\pi\times 6^2}=\frac{360\pi}{36\pi}=10$

The central angle is degrees.

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

A sector in a circle with a radius of has an area of $2\pi$ . In degrees, what is the measurement of the central angle in the sector?

Possible Answers:

$34.8^\circ$

$28.8^\circ$

$32.6^\circ$

$30.2^\circ$

Correct answer:

$28.8^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 2\pi}{\pi\times 5^2}=\frac{720\pi}{25\pi}=28.8$

The central angle is 28.8 degrees.

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Example Question #7 : How To Find The Angle Of A Sector

A sector in a circle with a radius of has an area of $60\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$305.5^\circ$

$355.5^\circ$

$332.5^\circ$

$337.5^\circ$

Correct answer:

$337.5^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 60\pi}{\pi\times 8^2}=\frac{21600\pi}{64\pi}=337.5$

The central angle is 337.5 degrees.

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Example Question #11 : How To Find The Angle Of A Sector

A sector in a circle with a radius of has an area of $3\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$11.2^\circ$

$10.8^\circ$

$11^\circ$

$11.4^\circ$

Correct answer:

$10.8^\circ$

Explanation:

Recall how to find the area of a sector:

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

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

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

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 10^2}=\frac{1080\pi}{100\pi}=10.8$

The central angle is 10.8 degrees.

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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

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

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

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

Example Question #4 : How To Find The Angle Of A Sector

Example Question #5 : How To Find The Angle Of A Sector

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

Example Question #7 : How To Find The Angle Of A Sector

Example Question #11 : How To Find The Angle Of A Sector

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