Question

if a circle has a central angle of 85 degrees and a radius of 16 millimeters how long is the arc formed by that angle? leave your answer in terms of pi

Answer 1

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

\[
\text{Arc Length} = \frac{\theta}{360} \times 2\pi r
\]

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

Given that the central angle \( \theta = 85\) degrees and the radius \( r = 16\) millimeters, we can substitute these values into the formula:

\[
\text{Arc Length} = \frac{85}{360} \times 2\pi \times 16
\]

First, simplify \( \frac{85}{360} \):

\[
\frac{85}{360} = \frac{17}{72}
\]

Now, substitute this back into the formula for arc length:

\[
\text{Arc Length} = \frac{17}{72} \times 2\pi \times 16
\]

Calculating \( 2\pi \times 16 \):

\[
2\pi \times 16 = 32\pi
\]

Now, substituting this into the equation gives:

\[
\text{Arc Length} = \frac{17}{72} \times 32\pi
\]

Now simplify \( \frac{17 \times 32\pi}{72} \):

\[
\text{Arc Length} = \frac{544\pi}{72}
\]

Now simplify \( \frac{544}{72} \):

Dividing both the numerator and denominator by 8:

\[
\frac{544 \div 8}{72 \div 8} = \frac{68}{9}
\]

Therefore, the length of the arc is:

\[
\text{Arc Length} = \frac{68\pi}{9} \text{ millimeters}
\]

Thus, the final answer for the length of the arc, in terms of \(\pi\), is:

\[
\frac{68\pi}{9} \text{ millimeters}
\]