Question

the circle has center o. its radius is 2 cm, and the central angle α measures 150°. what is the area of the shaded region? give the exact answer in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta$ is the central - angle of the sector and $r$ is the radius of the circle.

Step2: Identify the values of $\theta$ and $r$

We are given that $r = 2$ cm and $\theta=150^{\circ}$.

Step3: Substitute the values into the formula

$A=\frac{150^{\circ}}{360^{\circ}}\times\pi\times(2)^{2}$.
First, simplify $\frac{150}{360}=\frac{5}{12}$, and $(2)^{2}=4$. Then $A=\frac{5}{12}\times\pi\times4$.
$A=\frac{5\pi}{3}$ $cm^{2}$.

Answer:

$\frac{5\pi}{3}$ $cm^{2}$