To solve this problem, we need to find the area of the segment of a circle cut off by one side of an inscribed equilateral triangle.
1. Determine the central angle:
Since the triangle is equilateral, it divides the circle into three equal sectors. Each sector has a central angle of [tex]$60^\circ$[/tex] or [tex]\(\frac{\pi}{3}\)[/tex] radians.
2. Calculate the area of the sector:
The area [tex]\(A_{\text{sector}}\)[/tex] of a sector with a central angle [tex]\(\theta\)[/tex] in a circle of radius [tex]\(r\)[/tex] is given by:
[tex]\[
A_{\text{sector}} = \frac{1}{2} r^2 \theta
\][/tex]
Plugging in the given values ([tex]\(\theta = \frac{\pi}{3}\)[/tex] and [tex]\(r = 6\)[/tex]):
[tex]\[
A_{\text{sector}} = \frac{1}{2} \times 6^2 \times \frac{\pi}{3} = 18.84955592153876 \text{ square inches}
\][/tex]
3. Calculate the area of the equilateral triangle:
The area [tex]\(A_{\text{triangle}}\)[/tex] of an equilateral triangle with side length [tex]\(s\)[/tex] can be found using:
[tex]\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2
\][/tex]
The side length [tex]\(s\)[/tex] of the triangle can be found using the formula [tex]\(s = r\sqrt{3}\)[/tex]. Thus:
[tex]\[
s = 6 \sqrt{3}
\][/tex]
Therefore, the area becomes:
[tex]\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} (6 \sqrt{3})^2 = 46.76537180435968 \text{ square inches}
\][/tex]
4. Calculate the area of the segment:
The segment area is the area of the sector minus the area of the triangle:
[tex]\[
A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = 18.84955592153876 - 46.76537180435968 = -27.915815882820922
\][/tex]
Thus, the area of the segment cut off by one side of the equilateral triangle, expressed in exact form, is:
[tex]\[
A = (\pi - \sqrt{27.915815882820922}) \; \text{inches}^2
\][/tex]
