Sure! Let's go through the steps to find the area of a sector of a circle when the radius and the angle of the sector are given.
1. First, we need to know the formula for the area of a circle. The area \(A\) of a circle with radius \(r\) is given by:
[tex]\[
A = \pi r^2
\][/tex]
2. Given the radius \(r = 6\) inches, we can substitute this value into the formula:
[tex]\[
A = \pi \cdot 6^2 = \pi \cdot 36
\][/tex]
Thus, the area of the entire circle is:
[tex]\[
A = 36\pi \text{ square inches}
\][/tex]
3. Next, we need to find the area of the sector. The area of a sector of a circle is a fraction of the area of the entire circle. The fraction is determined by the ratio of the central angle \(\theta\) of the sector to \(360^\circ\) (since \(360^\circ\) corresponds to the entire circle).
4. In this case, the angle \(\theta\) is \(90^\circ\). Thus, the fraction of the circle that the sector represents is:
[tex]\[
\frac{\theta}{360} = \frac{90}{360} = \frac{1}{4}
\][/tex]
5. Therefore, the area of the sector is \(\frac{1}{4}\) of the area of the whole circle. We multiply the area of the circle by this fraction:
[tex]\[
\text{Area of the sector} = \left(\frac{1}{4}\right) \cdot 36\pi = 9\pi \text{ square inches}
\][/tex]
So, the exact area of the sector bounded by a \(90^\circ\) arc is:
[tex]\[
9\pi \text{ square inches}
\][/tex]
