Question

Find the area of the sector for a circle with a radius of 13 mm, and a central angle of 285°. Leave your answer in terms of π.(1 point) Responses 3211π24 mm2 3211π24 mm2 40π3 mm2 40π3 mm2 1859π12 mm2 1859π12 mm2 26π mm2 26π mm2

Answer 1

To find the area of a sector of a circle, you can use the formula:

\[
\text{Area} = \frac{\theta}{360} \times \pi r^2
\]

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

Given:
- Radius \(r = 13\) mm
- Central angle \(\theta = 285^\circ\)

Now, substitute the values into the formula:

\[
\text{Area} = \frac{285}{360} \times \pi (13)^2
\]

First, calculate \(13^2\):

\[
13^2 = 169
\]

Now substitute back into the area formula:

\[
\text{Area} = \frac{285}{360} \times \pi \times 169
\]

Now, simplify \(\frac{285}{360}\):

\[
\frac{285}{360} = \frac{285 \div 15}{360 \div 15} = \frac{19}{24}
\]

Now, substitute this back into the area formula:

\[
\text{Area} = \frac{19}{24} \times \pi \times 169
\]

Calculate \(19 \times 169\):

\[
19 \times 169 = 3211
\]

So we have:

\[
\text{Area} = \frac{3211}{24} \pi
\]

Thus, the area of the sector is:

\[
\frac{3211\pi}{24} \text{ mm}^2
\]

The correct answer is:

\(\frac{3211\pi}{24} \text{ mm}^2\)