Question

The radius of a circle is 10 inches. What is the area of a sector bounded by a 126° arc?
126°
r=10 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:

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

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

In your case, the angle \( \theta \) is 126° and the radius \( r \) is 10 inches.

Plugging in the values:

\[
A = \frac{126}{360} \times \pi \times (10)^2
\]

Now we simplify the calculations step by step:

1. Calculate \( (10)^2 \):
\[
(10)^2 = 100
\]

2. Now substitute this back:
\[
A = \frac{126}{360} \times \pi \times 100
\]

3. Simplify \( \frac{126}{360} \):
Both the numerator and the denominator can be divided by 18:
\[
\frac{126 \div 18}{360 \div 18} = \frac{7}{20}
\]

4. Now substitute this simplification:
\[
A = \frac{7}{20} \times \pi \times 100
\]

5. Simplify further:
\[
A = \frac{700}{20} \times \pi = 35\pi
\]

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

\[
\boxed{35\pi} \text{ square inches}
\]