Question

The radius of a circle is 2 feet. What is the area of a sector bounded by a 126° arc?
126°
r=2 ft
Give the exact answer in simplest form.

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 of the sector in degrees,
- \(r\) is the radius of the circle.

For this problem:
- \(\theta = 126^\circ\)
- \(r = 2 \text{ ft}\)

Now we can substitute these values into the formula:

\[
\text{Area of sector} = \frac{126}{360} \times \pi (2^2)
\]

First, calculate \(2^2\):

\[
2^2 = 4
\]

Now plug this back into the formula:

\[
\text{Area of sector} = \frac{126}{360} \times \pi \times 4
\]

Simplify \(\frac{126}{360}\):

\[
\frac{126}{360} = \frac{7}{20} \quad \text{(both numerator and denominator can be divided by 18)}
\]

Now substitute this back:

\[
\text{Area of sector} = \frac{7}{20} \times \pi \times 4
\]

Multiply:

\[
\text{Area of sector} = \frac{7 \times 4}{20} \pi = \frac{28}{20} \pi
\]

Simplify \(\frac{28}{20}\):

\[
\frac{28}{20} = \frac{7}{5}
\]

Thus, the area of the sector is:

\[
\text{Area of sector} = \frac{7}{5} \pi
\]

So, the exact area of the sector bounded by a 126° arc is:

\[
\boxed{\frac{7}{5} \pi} \text{ square feet}
\]