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Intermediate Geometry : How to find the angle of a sector

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Example Questions

Example Question #21 : Circles

Circle

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Possible Answers:

cannot be determined

100

Correct answer:

Explanation:

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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

What is the angle of a sector of area $\displaystyle 45$ in^2 $\displaystyle in^2$ on a circle having a radius of $15\:in$ $\displaystyle 15\:in$ ?

Possible Answers:

$3.00^{\circ}$ $\displaystyle 3.00^{\circ}$

$0.06^{\circ}$ $\displaystyle 0.06^{\circ}$

$72.00^{\circ}$ $\displaystyle 72.00^{\circ}$

$15.22^{\circ}$ $\displaystyle 15.22^{\circ}$

$22.92^{\circ}$ $\displaystyle 22.92^{\circ}$

Correct answer:

$22.92^{\circ}$ $\displaystyle 22.92^{\circ}$

Explanation:

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

$A=\pi r^2$ $\displaystyle A=\pi r^2$

For your data, this is:

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

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

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

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

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

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

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Example Question #521 : Plane Geometry

What is the angle of a sector that has an arc length of 13.5 $\displaystyle 13.5$ $\displaystyle in$ on a circle of diameter $\displaystyle 12$ $\displaystyle in$ ?

Possible Answers:

$14.24^{\circ}$ $\displaystyle 14.24^{\circ}$

$35.81^{\circ}$ $\displaystyle 35.81^{\circ}$

$128.92^{\circ}$ $\displaystyle 128.92^{\circ}$

$194.14^{\circ}$ $\displaystyle 194.14^{\circ}$

$10.74^{\circ}$ $\displaystyle 10.74^{\circ}$

Correct answer:

$128.92^{\circ}$ $\displaystyle 128.92^{\circ}$

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:

$C=\pi d$ $\displaystyle C=\pi d$

For your data, this means,

$C=12\pi$ $\displaystyle C=12\pi$

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

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

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

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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 $4\pi$ $\displaystyle 4\pi$ with a given radius of $\displaystyle 4$ ?

Possible Answers:

$360^{\circ}$ $\displaystyle 360^{\circ}$

$45^{\circ}$ $\displaystyle 45^{\circ}$

$60^{\circ}$ $\displaystyle 60^{\circ}$

$0^{\circ}$ $\displaystyle 0^{\circ}$

$90^{\circ}$ $\displaystyle 90^{\circ}$

Correct answer:

$90^{\circ}$ $\displaystyle 90^{\circ}$

Explanation:

Write the formula for the area of a circular sector.

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

Substitute the given information and solve for theta:

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

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

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

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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 $\small 12$ $\displaystyle \small 12$ meters is $\small \small 4\pi$ $\displaystyle \small \small 4\pi$ meters. Find the measure of the central angle of the sector.

Possible Answers:

$\small 30^\circ$ $\displaystyle \small 30^\circ$

$\small 45^\circ$ $\displaystyle \small 45^\circ$

$\small 15^\circ$ $\displaystyle \small 15^\circ$

$\small 135^\circ$ $\displaystyle \small 135^\circ$

$\small 60^\circ$ $\displaystyle \small 60^\circ$

Correct answer:

$\small 60^\circ$ $\displaystyle \small 60^\circ$

Explanation:

The formula for the length of an arc is

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

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

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

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

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

$\small \small m=60$ $\displaystyle \small \small m=60$

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

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Example Question #101 : Sectors

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

Possible Answers:

$60^\circ$ $\displaystyle 60^\circ$

$45^\circ$ $\displaystyle 45^\circ$

$30^\circ$ $\displaystyle 30^\circ$

$15^\circ$ $\displaystyle 15^\circ$

Correct answer:

$30^\circ$ $\displaystyle 30^\circ$

Explanation:

Recall how to find the area of a sector:

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

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

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

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

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

The central angle is $\displaystyle 30$ degrees.

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Example Question #103 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 4$ has an area of $12\pi$ $\displaystyle 12\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$255^\circ$ $\displaystyle 255^\circ$

$285^\circ$ $\displaystyle 285^\circ$

$270^\circ$ $\displaystyle 270^\circ$

$240^\circ$ $\displaystyle 240^\circ$

Correct answer:

$270^\circ$ $\displaystyle 270^\circ$

Explanation:

Recall how to find the area of a sector:

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

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

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

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

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

The central angle is 270 $\displaystyle 270$ 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 $\displaystyle 6$ has an area of $\pi$ $\displaystyle \pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$5^\circ$ $\displaystyle 5^\circ$

$20^\circ$ $\displaystyle 20^\circ$

$10^\circ$ $\displaystyle 10^\circ$

$15^\circ$ $\displaystyle 15^\circ$

Correct answer:

$10^\circ$ $\displaystyle 10^\circ$

Explanation:

Recall how to find the area of a sector:

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

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

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

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

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

The central angle is $\displaystyle 10$ 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 $\displaystyle 5$ has an area of $2\pi$ $\displaystyle 2\pi$ . In degrees, what is the measurement of the central angle in the sector?

Possible Answers:

$34.8^\circ$ $\displaystyle 34.8^\circ$

$28.8^\circ$ $\displaystyle 28.8^\circ$

$32.6^\circ$ $\displaystyle 32.6^\circ$

$30.2^\circ$ $\displaystyle 30.2^\circ$

Correct answer:

$28.8^\circ$ $\displaystyle 28.8^\circ$

Explanation:

Recall how to find the area of a sector:

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

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

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

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

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

The central angle is 28.8 $\displaystyle 28.8$ 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 $\displaystyle 8$ has an area of $60\pi$ $\displaystyle 60\pi$ . In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$355.5^\circ$ $\displaystyle 355.5^\circ$

$305.5^\circ$ $\displaystyle 305.5^\circ$

$332.5^\circ$ $\displaystyle 332.5^\circ$

$337.5^\circ$ $\displaystyle 337.5^\circ$

Correct answer:

$337.5^\circ$ $\displaystyle 337.5^\circ$

Explanation:

Recall how to find the area of a sector:

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

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

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

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

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

The central angle is 337.5 $\displaystyle 337.5$ degrees.

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Intermediate Geometry : How to find the angle of a sector

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : Circles

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

Example Question #521 : Plane Geometry

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

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

Example Question #101 : Sectors

Example Question #103 : Plane Geometry

Example Question #1 : 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

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