Question

choose the expression that correctly represents the conversion of degrees to radians.
a. radians = degrees / 180
b. radians = degrees × π / 180
c. radians = 180 / degrees
d. radians = degrees / π
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 radians
c. π/2 radians
d. 2π/3 radians

Explanation:

Step1: Degree - radian conversion formula

The formula for converting degrees to radians is \( \text{Radians}=\frac{\text{Degrees}\times\pi}{180} \).

Step2: Area of a sector formula

The area of a sector \( A=\frac{1}{2}r^{2}\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. The area of a full - circle \( A_{circle}=\pi r^{2} \). If the area of the sector \( A = \frac{1}{2}A_{circle}\), then \(\frac{1}{2}r^{2}\theta=\frac{1}{2}\pi r^{2}\). Canceling out \(\frac{1}{2}r^{2}\) from both sides gives \(\theta=\pi\) radians.

Answer:

For the conversion formula: B. Radians \(=\frac{\text{Degrees}\times\pi}{180}\)
For the central - angle problem: B. \(\pi\) radians