Question

The radius of a circle is 4 inches. What is the area of a sector bounded by a 90° 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 central angle of the sector in degrees, and \(r\) is the radius of the circle.

In this case, the radius \(r\) is 4 inches, and the angle \(\theta\) is 90°.

First, we calculate the area of the entire circle:

\[
\text{Area of the circle} = \pi r^2 = \pi (4)^2 = 16\pi \text{ square inches}
\]

Now, we substitute the values into the sector area formula:

\[
\text{Area of the sector} = \frac{90}{360} \times 16\pi
\]

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

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

Thus, we can calculate the area of the sector:

\[
\text{Area of the sector} = \frac{1}{4} \times 16\pi = 4\pi \text{ square inches}
\]

Therefore, the area of the sector bounded by a 90° arc is

\[
\boxed{4\pi} \text{ square inches.}
\]

Answer 2

The radius of a circle is 6 kilometers. What is the area of a sector bounded by a 135° arc?