Question

The radius of a circle is 6 kilometers. What is the area of a sector bounded by a 135° arc?
Give the exact answer in simplest form.

Answer 1

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

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

where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.

Given that the radius \(r = 6\) kilometers and the angle \(\theta = 135^\circ\), we substitute these values into the formula:

1. Calculate the area:

\[
\text{Area of the sector} = \frac{135^\circ}{360^\circ} \times \pi (6^2)
\]

2. Simplify the area calculation:

\[
= \frac{135}{360} \times \pi \times 36
\]

3. Reduce \(\frac{135}{360}\):

\[
= \frac{135 \div 45}{360 \div 45} = \frac{3}{8}
\]

4. Substitute back:

\[
= \frac{3}{8} \times \pi \times 36
\]

5. Calculate:

\[
= \frac{3 \times 36}{8} \pi = \frac{108}{8} \pi = \frac{27}{2} \pi
\]

Thus, the area of the sector bounded by a 135° arc is:

\[
\frac{27}{2} \pi \text{ square kilometers}
\]

This is the exact answer in simplest form.