Question

If a circle has a central angle of 125° and a radius of 27 millimeters, how long is the arc formed by that angle? Leave your answer in terms of π. (1 point) Responses 54π mm 54π mm 75π8 mm 75π8 mm 75π4 mm 75π4 mm 2025π8 mm

Answer 1

To find the length of the arc formed by a central angle in a circle, you can use the formula:

\[
\text{Arc length} = r \cdot \theta
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.

First, you need to convert the central angle from degrees to radians. The conversion from degrees to radians is done using the formula:

\[
\theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180}
\]

Given that the angle is \( 125^\circ \):

\[
\theta = 125 \times \frac{\pi}{180} = \frac{125\pi}{180} = \frac{25\pi}{36} \text{ radians}
\]

Now, plug this value into the arc length formula with the given radius \( r = 27 \) mm:

\[
\text{Arc length} = 27 \cdot \frac{25\pi}{36}
\]

Now, calculating the arc length:

\[
\text{Arc length} = \frac{27 \cdot 25\pi}{36} = \frac{675\pi}{36}
\]

To simplify \(\frac{675\pi}{36}\):

\[
\frac{675}{36} = \frac{675 \div 9}{36 \div 9} = \frac{75}{4}
\]

Thus, the arc length is:

\[
\frac{75\pi}{4} \text{ mm}
\]

So the final answer is:

\[
\boxed{\frac{75\pi}{4} \text{ mm}}
\]