Question

The radius of a circle is 3 miles. What is the area of a sector bounded by a 90° arc?
90°
r=3 mi
Give the exact answer in simplest form.

Answer 1

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

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

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

In this case, you have:
- \(\theta = 90^\circ\)
- \(r = 3 \, \text{miles}\)

Plugging in the values:

\[
\text{Area of the Sector} = \frac{90}{360} \times \pi (3)^2
\]

Simplifying \(\frac{90}{360}\):

\[
\frac{90}{360} = \frac{1}{4}
\]

Now, substituting back into the formula:

\[
\text{Area of the Sector} = \frac{1}{4} \times \pi \times 9
\]

\[
\text{Area of the Sector} = \frac{9\pi}{4}
\]

Thus, the exact area of the sector bounded by a 90° arc is:

\[
\frac{9\pi}{4} \, \text{square miles}
\]