Question

The radius of a circle is 6 inches. What is the area of a sector bounded by an 84° arc?
84°
r=6 in
Give the exact answer in simplest form.
square inches

Answer 1

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

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

where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.

In this case, the radius \(r\) is 6 inches, and the angle \(\theta\) is 84°.

First, we need to calculate \( \pi r^2 \):

\[
\pi r^2 = \pi (6^2) = \pi (36) = 36\pi
\]

Now we can calculate the area of the sector:

\[
\text{Area of sector} = \frac{84}{360} \times 36\pi
\]

Now simplify \(\frac{84}{360}\):

\[
\frac{84}{360} = \frac{7}{30}
\]

Then, substituting back into the area formula:

\[
\text{Area of sector} = \frac{7}{30} \times 36\pi = \frac{252\pi}{30}
\]

Now simplify \(\frac{252}{30}\):

\[
\frac{252}{30} = \frac{126}{15} = \frac{42}{5}
\]

Thus, the area of the sector is:

\[
\text{Area of sector} = \frac{42\pi}{5} \text{ square inches}
\]

So, the exact answer in simplest form is:

\[
\frac{42\pi}{5} \text{ square inches}
\]