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Intermediate Geometry : Sectors

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Example Questions

Example Question #1 : How To Find The Length Of An Arc

If the area of a circle is 1.44 $\pi$ , what is its circumference?

Possible Answers:

$2.4\pi$

$7.54\pi$

$1.2\pi$

$2.88\pi$

$1.44\pi$

Correct answer:

$2.4\pi$

Explanation:

The answer is $2.4 \pi$ .

Utilizing the formula for area of a circle $\left( \pi r^{2}\right)$ , you would plug in the answer for area as

$1.44\pi$ = $\pi r^{2}$

Divide both sides by $\pi$ .

Then square root both sides to get

1.2=r

Then plug in 1.2 for in the equation for circumference for a circle, $2\pi r$ . Thus

$C = 2.4\pi$

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Example Question #1 : How To Find The Length Of An Arc

Circle

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Possible Answers:

$\frac{\pi r}{16}$

$\frac{\pi r}{4}$

$\frac{\pi r}{2}$

$\pi r$

$\frac{\pi r}{8}$

Correct answer:

$\frac{\pi r}{4}$

Explanation:

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Example Question #73 : Circles

A sector of a circle with radius of feet has an area of $\small 36\pi$ square feet. Find the length of the arc of the sector.

Possible Answers:

$\small 3\pi\,ft.$

$\small 12\pi\,ft$

$\small 30\pi\,ft.$

$\small 6\pi\,ft.$

$\small 24\pi\,ft.$

Correct answer:

$\small 6\pi\,ft.$

Explanation:

We begin with the formula for the area of a sector

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

where $\small m$ is the measure of the central angle in degrees and $\small r$ is the radius.

Substituting what we know gives

$\small 36\pi=\frac{m}{360}\pi (12^2)$

$\small 36\pi=\frac{4\pi m}{10}$

$\small 360\pi=4\pi m$

$\small 90=m$

Therefore our central angle is $\small 90^\circ$

We then turn to our formula for arc length.

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

Substituting gives

$\small s=\frac{90}{360}(2\pi)(12)=\frac{1}{4}(24\pi)=6\pi$

Therefore, our arc length is $\small 6\pi\,ft.$

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Example Question #2 : How To Find The Length Of An Arc

Find the perimeter around the following semicircle.

Semicirc

Possible Answers:

$27\pi \ cm$

$9\pi \ cm$

$18\pi \ cm$

$18\pi +18\ cm$

$9\pi +18\ cm$

Correct answer:

$9\pi +18\ cm$

Explanation:

The answer is $9\pi + 18cm$ .

First, you would need to find the radius of the semi-circle. 18 divided by 2 results in 9 cm for the radius. Then you would take the formula for finding circumference $C=2\pi r$ and plug in 9=r to get

$C=18\pi$ .

Then you would divide that result by 2 to get $9\pi$ since it is a semicircle. Lastly you would add 18 cm to $9\pi$ because the perimeter is the sum of the semicircle and the diameter. Remember that they are not like terms.

If you chose $9\pi$ , you forgot to include the diameter.

If you chose $27\pi$ , you added and , but they are not like terms.

If you chose $18\pi + 18$ , remember that you only need half of the circumference.

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Example Question #4 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc if it has a measure of 120 degrees.

Possible Answers:

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

$\frac{1}{3}\pi$

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

$\pi$

Correct answer:

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

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:

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

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

$\text{Arc Length}=\frac{120}{360}\times 2\pi\times2=\frac{4}{3}\pi$

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Example Question #1 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc if it has a measurement of 240 degrees.

Possible Answers:

$2\pi$

$4\pi$

$8\pi$

$6\pi$

Correct answer:

$4\pi$

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:

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

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

$\text{Arc Length}=\frac{240}{360}\times 2\pi\times 3=4\pi$

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Example Question #1 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc that has a measurement of degrees.

Possible Answers:

$2\pi$

$\frac{1}{4}\pi$

$\pi$

$\frac{1}{2}\pi$

Correct answer:

$\pi$

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:

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

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

$\text{Arc Length}=\frac{45}{360}\times 2\pi\times 4=\pi$

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Example Question #2 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc if it has a measure of 150 degrees.

Possible Answers:

$\frac{10}{3}\pi$

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

$4\pi$

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

Correct answer:

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

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:

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

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

$\text{Arc Length}=\frac{150}{360}\times 2\pi\times 5=\frac{25}{6}\pi$

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Example Question #8 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc that has a measure of degrees.

Possible Answers:

$3\pi$

$2\pi$

$\frac{19}{6}\pi$

$\frac{17}{6}\pi$

Correct answer:

$\frac{19}{6}\pi$

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:

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

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

$\text{Arc Length}=\frac{95}{360}\times 2\pi\times 6=\frac{19}{6}\pi$

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Example Question #9 : How To Find The Length Of An Arc

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Possible Answers:

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

$\pi$

$\frac{1}{3}\pi$

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

Correct answer:

$\pi$

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:

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

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

$\text{Arc Length}=\frac{30}{360}\times 2\pi\times 6=\pi$

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Intermediate Geometry : Sectors

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #73 : Circles

Example Question #2 : How To Find The Length Of An Arc

Example Question #4 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #2 : How To Find The Length Of An Arc

Example Question #8 : How To Find The Length Of An Arc

Example Question #9 : How To Find The Length Of An Arc

All Intermediate Geometry Resources

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