Question

in circle s, the length of \\( \overparen{t u}=6 \pi \\) and \\( m \angle t s u=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 \(L=\frac{n}{360}\times2\pi r\), 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:
\(6\pi=\frac{120}{360}\times2\pi r\)
Simplify \(\frac{120}{360}=\frac{1}{3}\), so the equation becomes \(6\pi=\frac{1}{3}\times2\pi r\)
Multiply both sides by \(3\) to get \(18\pi = 2\pi r\)
Divide both sides by \(2\pi\): \(r = 9\)

Step2: Find the area of the sector

The formula for the area of a sector is \(A=\frac{n}{360}\times\pi r^{2}\)
Substitute \(n = 120^{\circ}\) and \(r = 9\) into the formula:
\(A=\frac{120}{360}\times\pi\times9^{2}\)
Simplify \(\frac{120}{360}=\frac{1}{3}\) and \(9^{2}=81\)
\(A=\frac{1}{3}\times\pi\times81\)
\(A = 27\pi\)

Answer:

\(27\pi\)