Question

what is the central angle of a sector in radians if its area is half the area of the entire circle? a. π/2 radians b. 2π radians c. 2π/3 radians d. π radians how many radians are there in a quarter of a circle? a. π/2 radians b. π/6 radians c. π/8 radians d. π/3 radians which of the following is a property of the radian measure? a. radians are primarily used for measuring volume. b. radians are independent of the circles radius. c. radians measure the area of a sector. d. radians directly relate the arc length to the radius.

Explanation:

Step1: Recall circle - sector area formula

The area of a circle is $A = \pi r^{2}$, and the area of a sector with central - angle $\theta$ (in radians) is $A_{s}=\frac{1}{2}r^{2}\theta$. If $A_{s}=\frac{1}{2}A$, then $\frac{1}{2}r^{2}\theta=\frac{1}{2}\pi r^{2}$. Canceling out $\frac{1}{2}r^{2}$ on both sides, we get $\theta = \pi$ radians.

Step2: Recall the radian measure of a full - circle

A full - circle has an angle of $2\pi$ radians. A quarter of a circle has an angle of $\frac{2\pi}{4}=\frac{\pi}{2}$ radians.

Step3: Recall the property of radian measure

The definition of radian measure is $\theta=\frac{s}{r}$, where $s$ is the arc length and $r$ is the radius of the circle. Radians directly relate the arc length to the radius.

Answer:

1. d. $\pi$ radians
2. a. $\frac{\pi}{2}$ radians
3. d. Radians directly relate the arc length to the radius.