How to find the length of an arc - Math

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Question

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

What is the length of arc A?

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Answer

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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Question

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

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Answer

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2_πr_, according to the formula for circumference.

length of arc AB = (5/18)(2_πr_) = (5/18)(2_π_(18)) = 10_π_.

Thus, the length of arc AB is 10_π_.

The total length of the perimeter is thus 36 + 10_π_.

The answer is 36 + 10_π_.

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Question

If a quarter of the area of a circle is $9\pi$ , then what is a quarter of the circumference of the circle?

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Answer

If a quarter of the area of a circle is $9\pi$ , then the area of the whole circle is $36\pi$ . This means that the radius of the circle is 6. The diameter is 12. Thus, the circumference of the circle is $12\pi$ . One fourth of the circumference is $3\pi$ .

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Question

Find the arc length of a sector that has an angle of 120 degrees and radius of 3.

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Answer

The equation for the arc length of a sector is $($\frac{x}{360}$)2\Pi r$ .

Substitute the given radius for and the given angle for to get the following equation:

$($\frac{120}{360}$)2\Pi(3)$

Simplify:

$=($\frac{1}{3}$)2\Pi (3)$

$=2\Pi$

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Question

$\textup{What is the length, in inches, of an arc with a central angle of }$

$72\textup{ degrees on a circle with a radius of 20 inches?}$

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Answer

$$\frac{\textup{length of arc}$}{\textup{circumference}}=\frac{\textup{central angle}$}{360}$

$\textup{Find circumference: }C=2\pi r = 2\pi 20 = 40\pi$

$$\frac{x}{40\pi }$= $\frac{72}{360}$::\textup{reduce: }$\frac{x}{40\pi }$=\frac{1}{5}$:\textup{ cross-multiply: }:5x=40\pi$

$x=8\pi$

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Question

Find the circumference of the following sector:

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Answer

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}$)+2(6m)$

$C= 3 \pi + 12\ m$

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Question

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

What is the length of arc A?

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Answer

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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Question

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

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Answer

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

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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Question

Find the perimeter around the following semicircle.

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Answer

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.

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Question

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.

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

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}$=\frac{60}{360}$\times 2\pi\times 8=\frac{2}{3}$\pi$

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Question

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

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Answer

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}$=\frac{200}{360}$\times 2\pi\times 9=10\pi$

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Question

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

Tap to reveal answer

Answer

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}$=\frac{40}{360}$\times 2\pi\times 10=\frac{20}{9}$\pi$

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Question

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

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Answer

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}$=\frac{130}{360}$\times 2\pi\times 12=\frac{26}{3}$\pi$

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