Question

in the figure below, circle o has a central angle of 120°. what is the area of the shaded sector of circle o in terms of r, the radius? \\(\frac{1}{3}\pi r^2\\) square units \\(\frac{2}{3}\pi r\\) square units \\(\frac{2}{3}\pi r^2\\) square units \\(\frac{4}{3}\pi r\\) square units

Explanation:

Step1: Find the measure of the shaded central angle

The total angle around a point is \(360^\circ\). The unshaded central angle is \(120^\circ\), so the shaded central angle \(\theta = 360^\circ - 120^\circ = 240^\circ\).

Step2: Recall the formula for the area of a sector

The formula for the area of a sector with central angle \(\theta\) (in degrees) and radius \(r\) is \(A=\frac{\theta}{360^\circ}\times\pi r^{2}\).

Step3: Substitute \(\theta = 240^\circ\) into the formula

Substitute \(\theta = 240^\circ\) into the formula: \(A=\frac{240^\circ}{360^\circ}\times\pi r^{2}\).
Simplify \(\frac{240}{360}=\frac{2}{3}\), so \(A = \frac{2}{3}\pi r^{2}\).

Answer:

\(\boldsymbol{\frac{2}{3}\pi r^{2}}\) square units (which corresponds to the option " \(\frac{2}{3}\pi r^{2}\) square units")