Question

Circle B is shown. Line segments A B and C B are radii. The length of A B is 6. Sector A B C is shaded.
The measure of central angle ABC is StartFraction pi Over 2 EndFraction radians.

What is the area of the shaded sector?

6Pi units squared
9Pi units squared
18Pi units squared
36Pi units squared

Answer 1

To find the area of the shaded sector \( ABC \), we can use the formula for the area of a sector given by:

\[
\text{Area} = \frac{1}{2} r^2 \theta
\]

where \( r \) is the radius and \( \theta \) is the measure of the central angle in radians.

1. We know the radius \( r = AB = 6 \).
2. The angle \( \theta = \frac{\pi}{2} \) radians.

Now, substituting the values into the formula:

\[
\text{Area} = \frac{1}{2} (6)^2 \left(\frac{\pi}{2}\right)
\]

Calculating the area step by step:

1. Calculate \( (6)^2 \):
\[
(6)^2 = 36
\]

2. Now calculate \( \frac{1}{2} \times 36 \):
\[
\frac{1}{2} \times 36 = 18
\]

3. Then multiply by \( \frac{\pi}{2} \):
\[
18 \times \frac{\pi}{2} = \frac{18\pi}{2} = 9\pi
\]

Therefore, the area of the shaded sector \( ABC \) is:

\[
\boxed{9\pi} \text{ units squared}
\]