Question

what is the area of the sector of a circle with a central angle of 168 degrees and a radius of 15 inches?

Answer 1

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

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

where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the circle.

Given:
- \(\theta = 168^\circ\)
- \(r = 15\) inches

Now, substitute these values into the formula:

\[
\text{Area of the sector} = \frac{168}{360} \times \pi \times (15)^2
\]

First, calculate \(15^2\):

\[
15^2 = 225
\]

Now plug that into the formula:

\[
\text{Area of the sector} = \frac{168}{360} \times \pi \times 225
\]

Calculating \(\frac{168}{360}\):

\[
\frac{168}{360} = \frac{14}{30} = \frac{7}{15}
\]

Now, substitute this back into the equation:

\[
\text{Area of the sector} = \frac{7}{15} \times \pi \times 225
\]

Next, calculate \(\frac{7}{15} \times 225\):

\[
\frac{7 \times 225}{15} = \frac{1575}{15} = 105
\]

Therefore, the area of the sector is:

\[
\text{Area of the sector} = 105\pi \text{ square inches}
\]

If you would like a numerical approximation, using \(\pi \approx 3.14\):

\[
105\pi \approx 105 \times 3.14 \approx 329.7 \text{ square inches}
\]

Thus, the area of the sector of the circle is \(105\pi\) square inches, or approximately \(329.7\) square inches.