How to find the length of an arc

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Math › How to find the length of an arc

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Find the length of the arc if the radius of a circle is and the measure of the central angle is degrees.

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

$4\pi$

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

$5\pi$

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:

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

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

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

Solve.

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

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

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

Explanation

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

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

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

$4\pi$

$2\pi$

$6\pi$

$8\pi$

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.

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

A giant clock has a minute hand four and one-half feet in length. Since noon, the tip of the minute hand has traveled $110 \pi$ feet. Which of the following is true of the time right now?

The time is between 12:00 midnight and 12:30 AM.

The time is between 11:30 PM and 12:00 midnight.

The time is between 11:00 PM and 11:30 PM.

The time is between 12:30 AM and 1:00 AM.

The time is between 1:00 AM and 1:30 AM.

Explanation

Every hour, the tip of the minute hand travels the circumference of a circle, which here is

$C = 2 \pi r = 2 \pi \times 4 \frac{1}{2} = 9 \pi$ feet.

The minute hand has traveled $110 \pi$ feet since noon, so it has traveled the circumference of the circle

$\frac{110 \pi}{9 \pi} = \frac{110 }{9 } =12 \frac{2}{9}$ times.

Since $12 \leq 12 \frac{2}{9} \leq 12 \frac{1}{2}$ , between 12 and $12 \frac{1}{2}$ hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

Find the length of an arc if the radius of the circle is and the measurement of the central angle is degrees.

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

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

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

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

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:

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

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

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

Solve.

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

Find the perimeter around the following semicircle.

Semicirc

$9\pi +18\ cm$

$9\pi \ cm$

$18\pi \ cm$

$27\pi \ cm$

$18\pi +18\ cm$

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 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.

Find the circumference of the following sector:

$3 \pi + 12\ m$

$3 \pi + 6\ m$

$3 \pi + 8\ m$

$2 \pi + 12\ m$

$2 \pi + 8\ m$

Explanation

The formula for the circumference of a sector is

$C = 2 \pi r (part) + 2r$ ,

where is the radius of the sector and is the fraction of the sector.

Plugging in our values, we get:

$C = 2 \pi (6m)(\frac{90}{360})+2(6m)$

$C= 3 \pi + 12\ m$

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

$\pi$

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

$2\pi$

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

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.

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

Acute triangle $\bigtriangleup ABC$ is inscribed in a circle. Which is the greater quantity?

(a) $m \overarc {ABC} - m \overarc {ACB}$

(b) $m \overarc {A B} - m \overarc {A C}$

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

Explanation

Examine the figure below, which shows $\bigtriangleup ABC$ inscribed in a circle.

Inscribed angle

By the Arc Addition Principle,

$m \overarc {ABC} = m \overarc {AB}+ m \overarc {BC}$

and

$m \overarc {ACB} = m \overarc {AC}+ m \overarc {BC}$

Consequently,

$m \overarc {ABC} - m \overarc {ACB}$

$= (m \overarc {AB}+ m \overarc {BC} ) - ( m \overarc {AC}+ m \overarc {BC})$

$= m \overarc {AB}+ m \overarc {BC} - m \overarc {AC}- m \overarc {BC}$

$= m \overarc {A B} - m \overarc {A C}$

The two quantities are equal.

Find the length of an arc if the radius of the circle is and the measurement of the central angle is degrees.

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

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

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

$8\pi$

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:

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

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

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

Solve.

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

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