Question

what is the area of the sector of a circle with θ = 75 degrees and a radius of 4 feet?

Answer 1

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

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

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

In this case, \(\theta = 75\) degrees and \(r = 4\) feet. Plugging in these values:

\[
\text{Area} = \frac{75}{360} \times \pi \times (4^2)
\]

Calculating \(4^2\):

\[
4^2 = 16
\]

Now substitute this back into the equation:

\[
\text{Area} = \frac{75}{360} \times \pi \times 16
\]

Calculating \(\frac{75}{360}\):

\[
\frac{75}{360} = \frac{5}{24}
\]

Now substitute that into the area formula:

\[
\text{Area} = \frac{5}{24} \times \pi \times 16
\]

Calculating \( \frac{5 \times 16}{24} \):

\[
\frac{5 \times 16}{24} = \frac{80}{24} = \frac{10}{3}
\]

Thus, the area becomes:

\[
\text{Area} = \frac{10}{3} \pi \text{ square feet}
\]

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

\[
\text{Area} \approx \frac{10}{3} \times 3.14 \approx 10.47 \text{ square feet}
\]

So, the area of the sector is:

\[
\frac{10}{3} \pi \text{ square feet} \quad \text{or approximately } 10.47 \text{ square feet.}
\]