Question

The radius of a circle is 6 feet. What is the area of a sector bounded by a 140° arc?
140°
r=6 ft
Give the exact answer in simplest form.

Answer 1

To find the area of a sector formed by a circle with a given angle and radius, you can use the formula:

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

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

In this case, the radius \(r = 6\) feet and the angle \(\theta = 140^\circ\). We can now substitute these values into the formula:

\[
\text{Area of the sector} = \frac{140}{360} \times \pi \times (6)^2
\]

Calculating the squared radius:

\[
(6)^2 = 36
\]

Now substituting this back into the equation:

\[
\text{Area of the sector} = \frac{140}{360} \times \pi \times 36
\]

Now simplify \(\frac{140}{360}\):

\[
\frac{140}{360} = \frac{7}{18}
\]

So the equation for the area becomes:

\[
\text{Area of the sector} = \frac{7}{18} \times \pi \times 36
\]

Now, multiplying:

\[
\text{Area of the sector} = \frac{7 \times 36}{18} \times \pi
\]

\[
36 \div 18 = 2
\]

Thus, it simplifies to:

\[
\text{Area of the sector} = 7 \times 2 \times \pi = 14\pi
\]

Thus, the exact area of the sector bounded by a \(140^\circ\) arc is:

\[
\boxed{14\pi} \text{ square feet.}
\]