Question

The radius of a circle is 3 miles. What is the area of a sector bounded by a 54° arc?
54°
r=3 mi
Give the exact answer in simplest form.

Answer 1

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

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

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

Given:
- The radius \( r = 3 \) miles
- The angle \( \theta = 54^\circ \)

Now, we substitute the values into the formula:

\[
\text{Area of Sector} = \frac{54^\circ}{360^\circ} \times \pi (3)^2
\]

Now calculate \( (3)^2 = 9 \):

\[
\text{Area of Sector} = \frac{54}{360} \times \pi \times 9
\]

Now simplify \( \frac{54}{360} \):

\[
\frac{54}{360} = \frac{3}{20}
\]

So, we have:

\[
\text{Area of Sector} = \frac{3}{20} \times \pi \times 9
\]

Now calculate \( 3 \times 9 = 27 \):

\[
\text{Area of Sector} = \frac{27}{20} \pi
\]

Therefore, the exact area of the sector bounded by a 54° arc is:

\[
\boxed{\frac{27}{20} \pi} \text{ square miles}
\]