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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #81 : Plane Geometry

The radius of a circle is $\displaystyle 8$. Find the length of an arc if it has a measure of $\displaystyle 60$ degrees.

Possible Answers:

$\frac{4}{3}\pi$ $\displaystyle \frac{4}{3}\pi$

$\frac{2}{3}\pi$ $\displaystyle \frac{2}{3}\pi$

$\frac{8}{3}\pi$ $\displaystyle \frac{8}{3}\pi$

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

Correct answer:

$\frac{2}{3}\pi$ $\displaystyle \frac{2}{3}\pi$

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{60}{360}\times 2\pi\times 8=\frac{2}{3}\pi$ $\displaystyle \text{Arc Length}=\frac{60}{360}\times 2\pi\times 8=\frac{2}{3}\pi$

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Example Question #81 : Sectors

The radius of a circle is $\displaystyle 9$. Find the length of an arc if it has a measure of 200 $\displaystyle 200$ degrees.

Possible Answers:

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

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

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

$15\pi$ $\displaystyle 15\pi$

Correct answer:

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

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{200}{360}\times 2\pi\times 9=10\pi$ $\displaystyle \text{Arc Length}=\frac{200}{360}\times 2\pi\times 9=10\pi$

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Example Question #81 : Intermediate Geometry

The radius of a circle is $\displaystyle 10$. Find the length of an arc if it has a measure of $\displaystyle 40$ degrees.

Possible Answers:

$\frac{20}{9}\pi$ $\displaystyle \frac{20}{9}\pi$

$\frac{2}{9}\pi$ $\displaystyle \frac{2}{9}\pi$

$\frac{22}{9}\pi$ $\displaystyle \frac{22}{9}\pi$

$\frac{25}{9}\pi$ $\displaystyle \frac{25}{9}\pi$

Correct answer:

$\frac{20}{9}\pi$ $\displaystyle \frac{20}{9}\pi$

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{40}{360}\times 2\pi\times 10=\frac{20}{9}\pi$ $\displaystyle \text{Arc Length}=\frac{40}{360}\times 2\pi\times 10=\frac{20}{9}\pi$

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Example Question #84 : Plane Geometry

The radius of a circle is $\displaystyle 12$. Find the length of an arc if it has a measure of 130 $\displaystyle 130$ degrees.

Possible Answers:

$\frac{26}{3}\pi$ $\displaystyle \frac{26}{3}\pi$

$\frac{22}{3}\pi$ $\displaystyle \frac{22}{3}\pi$

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

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

Correct answer:

$\frac{26}{3}\pi$ $\displaystyle \frac{26}{3}\pi$

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{130}{360}\times 2\pi\times 12=\frac{26}{3}\pi$ $\displaystyle \text{Arc Length}=\frac{130}{360}\times 2\pi\times 12=\frac{26}{3}\pi$

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Example Question #81 : Sectors

The radius of a circle is $\displaystyle 12$. Find the length of an arc if it has a measure of 180 $\displaystyle 180$ degrees.

Possible Answers:

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

$24\pi$ $\displaystyle 24\pi$

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

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

Correct answer:

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

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{180}{360}\times 2\pi\times 12=12\pi$ $\displaystyle \text{Arc Length}=\frac{180}{360}\times 2\pi\times 12=12\pi$

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Example Question #81 : Intermediate Geometry

The radius of a circle is $\displaystyle 1$. Find the length of an arc if it has a measure of 210 $\displaystyle 210$ degrees.

Possible Answers:

$\frac{4}{3}\pi$ $\displaystyle \frac{4}{3}\pi$

$\frac{7}{6}\pi$ $\displaystyle \frac{7}{6}\pi$

$\frac{11}{6}\pi$ $\displaystyle \frac{11}{6}\pi$

$\frac{13}{6}\pi$ $\displaystyle \frac{13}{6}\pi$

Correct answer:

$\frac{7}{6}\pi$ $\displaystyle \frac{7}{6}\pi$

Explanation:

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

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

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{210}{360}\times 2\pi\times 1=\frac{7}{6}\pi$ $\displaystyle \text{Arc Length}=\frac{210}{360}\times 2\pi\times 1=\frac{7}{6}\pi$

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Example Question #87 : Plane Geometry

Find the length of the arc if the radius of a circle is $\displaystyle 12$ and the measure of the central angle is $\displaystyle 55$ degrees.

Possible Answers:

$\frac{11}{3}\pi$ $\displaystyle \frac{11}{3}\pi$

$5\pi$ $\displaystyle 5\pi$

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

$\frac{13}{3}\pi$ $\displaystyle \frac{13}{3}\pi$

Correct answer:

$\frac{11}{3}\pi$ $\displaystyle \frac{11}{3}\pi$

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:

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

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

$\text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$ $\displaystyle \text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$

Solve.

$\text{Length of Arc}=\frac{11}{3}\pi$ $\displaystyle \text{Length of Arc}=\frac{11}{3}\pi$

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Example Question #88 : Plane Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 13$ and the measurement of the central angle is $\displaystyle 65$ degrees.

Possible Answers:

$\frac{169}{36}\pi$ $\displaystyle \frac{169}{36}\pi$

$\frac{19}{4}\pi$ $\displaystyle \frac{19}{4}\pi$

$\frac{37}{6}\pi$ $\displaystyle \frac{37}{6}\pi$

$\frac{110}{7}\pi$ $\displaystyle \frac{110}{7}\pi$

Correct answer:

$\frac{169}{36}\pi$ $\displaystyle \frac{169}{36}\pi$

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.

$\text{Length of Arc}=\frac{65}{360}\times 2\pi \times 13$ $\displaystyle \text{Length of Arc}=\frac{65}{360}\times 2\pi \times 13$

Solve.

$\text{Length of Arc}=\frac{169}{36}\pi$ $\displaystyle \text{Length of Arc}=\frac{169}{36}\pi$

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Example Question #81 : Sectors

Find the length of an arc if the radius of the circle is $\displaystyle 14$ and the measurement of the central angle is $\displaystyle 75$ degrees.

Possible Answers:

$\frac{13}{3}\pi$ $\displaystyle \frac{13}{3}\pi$

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

$\frac{25}{6}\pi$ $\displaystyle \frac{25}{6}\pi$

$\frac{35}{6}\pi$ $\displaystyle \frac{35}{6}\pi$

Correct answer:

$\frac{35}{6}\pi$ $\displaystyle \frac{35}{6}\pi$

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.

$\text{Length of Arc}=\frac{75}{360}\times 2\pi \times 14$ $\displaystyle \text{Length of Arc}=\frac{75}{360}\times 2\pi \times 14$

Solve.

$\text{Length of Arc}=\frac{35}{6}\pi$ $\displaystyle \text{Length of Arc}=\frac{35}{6}\pi$

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Example Question #81 : Plane Geometry

Find the length of an arc if the radius of the circle is $\displaystyle 15$ and the measurement of the central angle is $\displaystyle 80$ degrees.

Possible Answers:

$\frac{20}{3}\pi$ $\displaystyle \frac{20}{3}\pi$

$\frac{29}{3}\pi$ $\displaystyle \frac{29}{3}\pi$

$\frac{22}{3}\pi$ $\displaystyle \frac{22}{3}\pi$

$6\pi$ $\displaystyle 6\pi$

Correct answer:

$\frac{20}{3}\pi$ $\displaystyle \frac{20}{3}\pi$

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.

$\text{Length of Arc}=\frac{80}{360}\times 2\pi \times 15$ $\displaystyle \text{Length of Arc}=\frac{80}{360}\times 2\pi \times 15$

Solve.

$\text{Length of Arc}=\frac{20}{3}\pi$ $\displaystyle \text{Length of Arc}=\frac{20}{3}\pi$

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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #81 : Plane Geometry

Example Question #81 : Sectors

Example Question #81 : Intermediate Geometry

Example Question #84 : Plane Geometry

Example Question #81 : Sectors

Example Question #81 : Intermediate Geometry

Example Question #87 : Plane Geometry

Example Question #88 : Plane Geometry

Example Question #81 : Sectors

Example Question #81 : Plane Geometry

All Intermediate Geometry Resources

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