Question

if the central angle of a sector is 3 radians and the radius is 6 inches, what is the sector area?
a. 27π square inches
b. 18 square inches
c. 12π square inches
d. 54 square inches

if a central angle measures π/4 radians and the radius of the circle is 10 cm, what is the arc length?
a. 2π/3 cm
b. 5π/2 cm
c. 10π cm
d. 5π cm

what is the sector area of a circle with radius 12 cm if the central angle is π/6 radians?
a. 12π square cm
b. 24π square cm
c. 18π square cm
d. 6π square cm

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians.

Step2: Substitute values for the first question

Given $r = 6$ inches and $\theta=3$ radians. Substitute into the formula: $A=\frac{1}{2}\times6^{2}\times3=\frac{1}{2}\times36\times3 = 54$ square inches.

Step3: Recall arc - length formula

The formula for the arc - length of a circle is $s = r\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians.

Step4: Substitute values for the second question

Given $r = 10$ cm and $\theta=\frac{\pi}{4}$ radians. Substitute into the formula: $s=10\times\frac{\pi}{4}=\frac{5\pi}{2}$ cm.

Step5: Substitute values for the third question

Given $r = 12$ cm and $\theta=\frac{\pi}{6}$ radians. Using the sector - area formula $A=\frac{1}{2}r^{2}\theta$, we have $A=\frac{1}{2}\times12^{2}\times\frac{\pi}{6}=\frac{1}{2}\times144\times\frac{\pi}{6}=12\pi$ square cm.

Answer:

1. d. 54 square inches
2. b. $\frac{5\pi}{2}$ cm
3. a. $12\pi$ square cm