Pre-Calculus - Find the Length of the Arc of a Circle Using Radians

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

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

$\frac{5\pi}{4}$

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

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Find the arc length of a circle with circumference $10\pi$ that goes from to $135^\circ$ around its center.

$A=\frac{15\pi}{4}$

$A=\frac{25\pi}{4}$

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

Explanation

Remember that arc length of a circle is given by:

$A=\theta r$ , where $\theta$ is angle made by connecting the radius to the two ends of the arc, given in radians. is the radius.

To get $\theta$ , we convert from degrees to radians.

$\theta = 135deg\cdot\frac{\pi rad}{180deg}=\frac{3\pi}{4}$

To get the radius we convert from the circumference

$r=\frac{C}{2\pi}=\frac{10\pi}{2\pi}=5$

$A=\frac{3\pi}{4}\cdot5= \frac{15\pi}{4}$

If circle has a radius of ${}\frac{5}{\pi}$ fathoms and sector has a central angle of $\pi$ . What is the measure of arc ?

$5\pi fathoms$

$\frac{5}{\pi} fathoms$

Explanation

To find the length of an arc, use the following formula:

$\frac{\theta }{2\pi}\cdot 2\pi r=arc_{length}$

We are given the radius, as well as our central angle, so plug in what we know and simplify

$\frac{\pi }{2\pi}\cdot 2\pi \cdot \frac{5}{\pi}=\frac{1}{2}\cdot 2\cdot 10=5$

So our answer is 5 fathoms

If a circle has a circumference of 310 kilometers, find the length of the arc associated with a central angle of $\frac{3\pi}{4}$ radians.

Explanation

If a circle has a circumference of 310 kilometers, find the length of the arc associated with a central angle of $\frac{3\pi}{4}$ radians

Recall that arc length can be found via the following:

$Arc=\frac{\measuredangle }{2\pi}*2\pi r$

Upon closer examination, we see that the formula is really two parts. The first part gives us the fractional area of the circle we care about. The second part is simply the circumference. This means that if we multiply the whole circumference by the fraction that we care about, we will just get the arc length of the portion we care about.

So, plug it in and go!

$Arc=\frac{\frac{3\pi}{4}}{2\pi}* (310km)$

Looks messy, but we'll simplify it.

$Arc=\frac{\frac{3\pi}{4}}{ 2\pi}* (310km)=\frac{3}{8}*310km$

$Arc=\frac{3}{8}*310km\approx116km$

So our answer is 116km.

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If the length of is and $\angle O=\frac{\pi}{5}$ , which of the following is closest to the length of minor arc ?

Explanation

The formula for arc length of a circle, given in radians, is $s=\theta \cdot r$ , where is the arc length.

Then, plug in the givens: $s=\frac{\pi}{5}\cdot 3$ and simplify: .

Then round to the appropriate placement: .

Therefore, the arc length of is .

Find the length of a circular arc if the radius is , with an angle of $\frac{\pi}{2}$ .

$\pi$

$\frac{\pi}{4}$

$2\pi$

$\frac{\pi}{2}$

Explanation

Write the formula to find the length of an arc given the angle in radians.

$s=r\theta$

Substitute the radius and the angle in order to find the length of the arc.

$s=2\left(\frac{\pi}{2}\right)=\pi$

When CERN was designing its Large Hadron Collider, they needed to run a cable from the control house to a spot $\frac{\pi}{3}$ radiansaround the circle. If the Large Hadron Colider has a radius of , what length of cable will be necessary to reach the sensor?

Explanation

When CERN was designing its Large Hadron Collider, they needed to run a cable from the control house to a spot $\frac{\pi}{3}$ radians around the circle. If the Large Hadron Colider has a radius of , what length of cable will be necessary to reach the sensor?

The formula for length of an arc is as follows:

$Arc=\frac{Central\angle }{2\pi}*2\pi r$

Thus, to find the measure of the central angle, what we are really doing is multiplying the total circumference by the fractional part of the circle we are interested in. In this case, we already have the radius and the central angle, we just need to plug and chug.

$Arc=\frac{\frac{\pi}{3}}{2\pi}*2\pi*4.29=\frac{\pi}{3}*4.29=4.49\approx4.5:km$