Question 740·Medium·Math

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A circle has a radius of 6 centimeters. What is the length, in centimeters, of the arc that subtends a central angle of 120 degrees?

An arc length is a distance along part of a circle. A central angle given in degrees tells you what fraction of the full $360^\circ$ turn the arc covers. Find the circle’s circumference from its radius, then take that fraction of it; Desmos can handle the number multiplying $\pi$ . Don’t give the whole circumference when the question asks for only part.

Hints

Hint 1
A full turn is $360^\circ$ . The central angle selects a fraction of that turn, and its arc takes up the same fraction of the circle’s edge. What fraction does the given angle select?
Hint 2
The distance around a whole circle is its circumference, $C=2\pi r$ , where $r$ is the radius. The arc is part of that distance, so use the angle’s fraction on the circumference.

Step-by-step

Take a fraction of the circumference
Step 1
Find the fraction of a full turn
A full turn is $360^\circ$ , so the given $120^\circ$ central angle selects one-third of the circle: $\frac{120}{360}=\frac13$ . That fraction describes a share of the circle, not a length in centimeters.
Step 2
Identify the whole circumference
The circumference $C$ is the distance around the whole circle. Substitute the given radius $r=6$ into $C=2\pi r$ : $C=2\pi(6)$ .
Step 3
Write the arc as part of the circumference
The arc length $s$ is a distance along the circle’s edge. The arc covers one-third of that edge, so take one-third of the circumference, not one-third of the circle’s area: $s=\frac13\bigl(2\pi(6)\bigr)$ .
Step 4
Calculate the length
Type $\frac13\cdot2\cdot6$ in Desmos to find the number multiplying $\pi$ . Desmos displays $4$ . Keep $\pi$ exact, so the arc is $4\pi$ centimeters long. Choice C.

Question 740·Medium·Math

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