Arc AB is $\frac{1}{6}$ of the circumference of a circle. What is the radian measure of the central angle?
A. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{2 \pi}{3}$
D. $\frac{5 \pi}{6}$
Answer (2)
The circumference of a circle is 2 π radians.
The arc AB is 6 1 of the circumference.
Calculate the central angle by multiplying 6 1 with 2 π .
The radian measure of the central angle is 3 π .
Explanation
Problem Analysis The problem states that arc AB is 6 1 of the circumference of a circle. We need to find the radian measure of the central angle subtended by this arc.
Finding the central angle The circumference of a circle is 2 π radians. Since arc AB is 6 1 of the circumference, the central angle is 6 1 of 2 π radians.
Calculating the radian measure To find the radian measure of the central angle, we multiply 6 1 by 2 π :
6 1 × 2 π = 6 2 π = 3 π Therefore, the radian measure of the central angle is 3 π .
Final Answer The radian measure of the central angle is 3 π .
Examples
Imagine you're cutting a pizza into 6 equal slices. The angle of each slice at the center of the pizza is the central angle. Since a full circle is 2 π radians, each slice would have a central angle of 3 π radians. This concept is also used in astronomy to measure the angular size of celestial objects or the angular distance between stars.
The radian measure of the central angle that corresponds to arc AB, which is 6 1 of the circumference of a circle, is 3 π . This is calculated by multiplying 6 1 by the total circumference of 2 π . Therefore, the answer is option B: 3 π .
;
