Question

Sector of circle diameter is 16ft with a angle of pie/2 radians

Answer 1

To find the area of a sector of a circle given its diameter and the angle in radians, we can use the following formula for the area of a sector:

\[
A = \frac{1}{2} \times r^2 \times \theta
\]

where:
- \( A \) is the area of the sector,
- \( r \) is the radius of the circle,
- \( \theta \) is the angle in radians.

First, we need to calculate the radius from the given diameter. The diameter is 16 ft, so the radius \( r \) is:

\[
r = \frac{\text{diameter}}{2} = \frac{16}{2} = 8 \text{ ft}
\]

Now we have the radius \( r = 8 \text{ ft} \) and the angle \( \theta = \frac{\pi}{2} \) radians.

Substituting the values into the area formula:

\[
A = \frac{1}{2} \times (8)^2 \times \frac{\pi}{2}
\]

Calculate \( (8)^2 \):

\[
(8)^2 = 64
\]

Now substitute that back into the equation:

\[
A = \frac{1}{2} \times 64 \times \frac{\pi}{2}
\]

Simplify the equation:

\[
A = \frac{64}{2} \times \frac{\pi}{2}
\]
\[
A = 32 \times \frac{\pi}{2}
\]
\[
A = 16\pi \text{ ft}^2
\]

Therefore, the area of the sector of the circle is \( 16\pi \) square feet, which is approximately:

\[
A \approx 16 \times 3.14159 \approx 50.27 \text{ ft}^2
\]

So, the area of the sector is \( 16\pi \) square feet or approximately \( 50.27 \) square feet.