Question

using the formula for the area of a sector, solve for the area of a sector with an angle of 135 degrees and a radius of 13 inches

Answer 1

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

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

where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius.

In this case, you have \( \theta = 135 \) degrees and \( r = 13 \) inches. Plugging in these values:

\[
A = \frac{135}{360} \times \pi \times (13)^2
\]

First, calculate \( (13)^2 \):

\[
(13)^2 = 169
\]

Now substitute this value back into the area formula:

\[
A = \frac{135}{360} \times \pi \times 169
\]

Next, simplify \( \frac{135}{360} \):

\[
\frac{135}{360} = \frac{135 \div 45}{360 \div 45} = \frac{3}{8}
\]

Now, substitute \( \frac{3}{8} \) into the formula:

\[
A = \frac{3}{8} \times \pi \times 169
\]

Now multiply:

\[
A = \frac{3 \times 169}{8} \times \pi = \frac{507}{8} \times \pi
\]

Finally, calculate \( \frac{507}{8} \):

\[
\frac{507}{8} = 63.375
\]

Therefore, the area of the sector is:

\[
A \approx 63.375\pi \text{ square inches} \approx 199.1 \text{ square inches (using } \pi \approx 3.14 \text{)}
\]

So the area of the sector is approximately:

\[
A \approx 199.1 \text{ square inches}
\]

Or you can express it as:

\[
A = \frac{507}{8} \pi \text{ square inches}
\]