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Intermediate Geometry : Sectors

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Example Questions

Example Question #91 : Intermediate Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 16$ and the measurement of the central angle is 100 $\displaystyle 100$ degrees.

Possible Answers:

$\frac{85}{9}\pi$ $\displaystyle \frac{85}{9}\pi$

$10\pi$ $\displaystyle 10\pi$

$9\pi$ $\displaystyle 9\pi$

$\frac{80}{9}\pi$ $\displaystyle \frac{80}{9}\pi$

Correct answer:

$\frac{80}{9}\pi$ $\displaystyle \frac{80}{9}\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$ $\displaystyle \text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{100}{360}\times 2\pi \times 16$ $\displaystyle \text{Length of Arc}=\frac{100}{360}\times 2\pi \times 16$

Solve.

$\text{Length of Arc}=\frac{80}{9}\pi$ $\displaystyle \text{Length of Arc}=\frac{80}{9}\pi$

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Example Question #91 : Plane Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 17$ and the measurement of the central angle is 120 $\displaystyle 120$ degrees.

Possible Answers:

$68\pi$ $\displaystyle 68\pi$

$66\pi$ $\displaystyle 66\pi$

$72\pi$ $\displaystyle 72\pi$

$70\pi$ $\displaystyle 70\pi$

Correct answer:

$68\pi$ $\displaystyle 68\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{120}{360}\times 2\pi \times 17$ $\displaystyle \text{Length of Arc}=\frac{120}{360}\times 2\pi \times 17$

Solve.

$\text{Length of Arc}=68\pi$ $\displaystyle \text{Length of Arc}=68\pi$

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Example Question #23 : How To Find The Length Of An Arc

Find the length of an arc if the radius of the circle is $\displaystyle 18$ and the measurement of the central angle is 125 $\displaystyle 125$ degrees.

Possible Answers:

$\frac{27}{2}\pi$ $\displaystyle \frac{27}{2}\pi$

$13\pi$ $\displaystyle 13\pi$

$\frac{25}{2}\pi$ $\displaystyle \frac{25}{2}\pi$

$12\pi$ $\displaystyle 12\pi$

Correct answer:

$\frac{25}{2}\pi$ $\displaystyle \frac{25}{2}\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{125}{360}\times 2\pi \times 18$ $\displaystyle \text{Length of Arc}=\frac{125}{360}\times 2\pi \times 18$

Solve.

$\text{Length of Arc}=\frac{25}{2}\pi$ $\displaystyle \text{Length of Arc}=\frac{25}{2}\pi$

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Example Question #24 : How To Find The Length Of An Arc

Find the length of an arc if the radius of the circle is $\displaystyle 18$ and the measurement of the central angle is 160 $\displaystyle 160$ degrees.

Possible Answers:

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

$12\pi$ $\displaystyle 12\pi$

$18\pi$ $\displaystyle 18\pi$

$20\pi$ $\displaystyle 20\pi$

Correct answer:

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

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{160}{360}\times 2\pi \times 18$ $\displaystyle \text{Length of Arc}=\frac{160}{360}\times 2\pi \times 18$

Solve.

$\text{Length of Arc}=16\pi$ $\displaystyle \text{Length of Arc}=16\pi$

Report an Error

Example Question #91 : Intermediate Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 20$ and the measurement of the central angle is 170 $\displaystyle 170$ degrees.

Possible Answers:

$\frac{170}{9}\pi$ $\displaystyle \frac{170}{9}\pi$

$\frac{185}{9}\pi$ $\displaystyle \frac{185}{9}\pi$

$20\pi$ $\displaystyle 20\pi$

$\frac{175}{9}\pi$ $\displaystyle \frac{175}{9}\pi$

Correct answer:

$\frac{170}{9}\pi$ $\displaystyle \frac{170}{9}\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{170}{360}\times 2\pi \times 20$ $\displaystyle \text{Length of Arc}=\frac{170}{360}\times 2\pi \times 20$

Solve.

$\text{Length of Arc}=\frac{170}{9}\pi$ $\displaystyle \text{Length of Arc}=\frac{170}{9}\pi$

Report an Error

Example Question #91 : Plane Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 21$ and the measurement of the central angle is 140 $\displaystyle 140$ degrees.

Possible Answers:

$\frac{49}{3}\pi$ $\displaystyle \frac{49}{3}\pi$

$17\pi$ $\displaystyle 17\pi$

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

$\frac{52}{3}\pi$ $\displaystyle \frac{52}{3}\pi$

Correct answer:

$\frac{49}{3}\pi$ $\displaystyle \frac{49}{3}\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{140}{360}\times 2\pi \times 21$ $\displaystyle \text{Length of Arc}=\frac{140}{360}\times 2\pi \times 21$

Solve.

$\text{Length of Arc}=\frac{49}{3}\pi$ $\displaystyle \text{Length of Arc}=\frac{49}{3}\pi$

Report an Error

Example Question #91 : Circles

Find the length of an arc if the radius of the circle is $\displaystyle 22$ and the measurement of the central angle is 200 $\displaystyle 200$ degrees.

Possible Answers:

$\frac{211}{9}\pi$ $\displaystyle \frac{211}{9}\pi$

$\frac{229}{9}\pi$ $\displaystyle \frac{229}{9}\pi$

$\frac{238}{9}\pi$ $\displaystyle \frac{238}{9}\pi$

$\frac{220}{9}\pi$ $\displaystyle \frac{220}{9}\pi$

Correct answer:

$\frac{220}{9}\pi$ $\displaystyle \frac{220}{9}\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{200}{360}\times 2\pi \times 22$ $\displaystyle \text{Length of Arc}=\frac{200}{360}\times 2\pi \times 22$

Solve.

$\text{Length of Arc}=\frac{220}{9}\pi$ $\displaystyle \text{Length of Arc}=\frac{220}{9}\pi$

Report an Error

Example Question #91 : Sectors

Find the length of an arc if the radius of the circle is $\displaystyle 24$ and the measurement of the central angle is 210 $\displaystyle 210$ degrees.

Possible Answers:

$28\pi$ $\displaystyle 28\pi$

$31\pi$ $\displaystyle 31\pi$

$30\pi$ $\displaystyle 30\pi$

$29\pi$ $\displaystyle 29\pi$

Correct answer:

$28\pi$ $\displaystyle 28\pi$

Explanation:

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{210}{360}\times 2\pi \times 24$ $\displaystyle \text{Length of Arc}=\frac{210}{360}\times 2\pi \times 24$

Solve.

$\text{Length of Arc}=28\pi$ $\displaystyle \text{Length of Arc}=28\pi$

Report an Error

Example Question #91 : Plane Geometry

In the figure below, $\text{chord AB}\parallel\text{chord CD}$ $\displaystyle \text{chord AB}\parallel\text{chord CD}$. If $\text{Arc AC}$ $\displaystyle \text{Arc AC}$ is $\displaystyle 18$ degrees, in degrees, what is the measure of $\text{Arc BD}$ $\displaystyle \text{Arc BD}$?

Possible Answers:

$\displaystyle 18$

$\displaystyle 22$

The measurement of $\text{Arc BD}$ $\displaystyle \text{Arc BD}$ cannot be determined with the information given.

$\displaystyle 16$

Correct answer:

$\displaystyle 18$

Explanation:

Recall that when chords are parallel, the arcs that are intercepted are congruent. Thus, $\text{Arc AC}=\text{Arc BD}$ $\displaystyle \text{Arc AC}=\text{Arc BD}$.

Then, $\text{Arc BD}$ $\displaystyle \text{Arc BD}$ must also be $\displaystyle 18$ degrees.

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

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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Intermediate Geometry : Sectors

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #91 : Intermediate Geometry

Example Question #91 : Plane Geometry

Example Question #23 : How To Find The Length Of An Arc

Example Question #24 : How To Find The Length Of An Arc

Example Question #91 : Intermediate Geometry

Example Question #91 : Plane Geometry

Example Question #91 : Circles

Example Question #91 : Sectors

Example Question #91 : Plane Geometry

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

All Intermediate Geometry Resources

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