Question

1. what is the central angle of a sector in radians if its area is half the area of the entire circle? a. 2π radians b. π/2 radians c. 2π/3 radians d. π radians

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector \(A=\frac{1}{2}r^{2}\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central - angle in radians. The area of a circle \(A_{c}=\pi r^{2}\).

Step2: Set up the equation based on the problem condition

We are given that the area of the sector is half the area of the circle. So, \(\frac{1}{2}r^{2}\theta=\frac{1}{2}(\pi r^{2})\).

Step3: Solve for \(\theta\)

Cancel out \(\frac{1}{2}r^{2}\) from both sides of the equation \(\frac{1}{2}r^{2}\theta=\frac{1}{2}(\pi r^{2})\). We get \(\theta = \pi\) radians.

Answer:

d. \(\pi\) radians