Question

in circle a, the length of \\( \overparen{bc}=6 \pi \\) and \\( m \angle b a c=120^{\circ} \\). find the area shaded below. express your answer as a fraction times \\( \pi \\).

Explanation:

Step1: Find the radius of the circle

The formula for the length of an arc is \(L=\frac{n\pi r}{180}\), where \(L\) is the arc length, \(n\) is the central angle in degrees, and \(r\) is the radius.
Given \(L = 6\pi\) and \(n=120^{\circ}\), we substitute into the formula:
\(6\pi=\frac{120\pi r}{180}\)
Cross - multiply: \(6\pi\times180 = 120\pi r\)
Divide both sides by \(120\pi\): \(r=\frac{6\pi\times180}{120\pi}=9\)

Step2: Find the area of the shaded region

The area of a circle is \(A=\pi r^{2}\), and the central angle of the shaded region is \(n = 360 - 120=240^{\circ}\)
The formula for the area of a sector is \(A_{sector}=\frac{n\pi r^{2}}{360}\)
Substitute \(n = 240\) and \(r = 9\) into the formula:
\(A_{sector}=\frac{240\pi\times9^{2}}{360}\)
\(A_{sector}=\frac{240\pi\times81}{360}\)
Simplify: \(A_{sector}=54\pi\)

Answer:

\(54\pi\)