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.
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