To find the exact area of the sector in terms of [tex]\(\pi\)[/tex], we can follow these steps:
1. Understand the components of the problem:
- The radius [tex]\(r\)[/tex] of the circle is 12 inches.
- The central angle [tex]\(\theta\)[/tex] is 60 degrees.
2. Formula for the area of a sector:
The area [tex]\(A\)[/tex] of a sector of a circle is given by the formula:
[tex]\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\][/tex]
where [tex]\(\theta\)[/tex] is the central angle in degrees, and [tex]\(r\)[/tex] is the radius of the circle.
3. Substitute the given values into the formula:
- [tex]\(\theta = 60^\circ\)[/tex]
- [tex]\(r = 12\)[/tex] inches
The formula becomes:
[tex]\[
A = \frac{60^\circ}{360^\circ} \times \pi \times (12)^2
\][/tex]
4. Simplify the expression step by step:
- First, simplify the fraction [tex]\(\frac{60}{360}\)[/tex]:
[tex]\[
\frac{60}{360} = \frac{1}{6}
\][/tex]
- Substitute this back into the equation:
[tex]\[
A = \frac{1}{6} \times \pi \times (12)^2
\][/tex]
- Calculate [tex]\(12^2\)[/tex]:
[tex]\[
12^2 = 144
\][/tex]
- Finally, simplify the multiplication:
[tex]\[
A = \frac{1}{6} \times \pi \times 144 = \frac{144}{6} \pi = 24 \pi
\][/tex]
So, the exact area of the sector in terms of [tex]\(\pi\)[/tex] is:
[tex]\[
24 \pi \ \text{in}^{2}
\][/tex]
This is the answer that should be entered in the box.
