Question 628·Easy·Math

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In a circle, a central angle measures 120 degrees. The circle has a circumference of 30.

What is the length of the minor arc intercepted by this angle?

A central angle tells you what fraction of a full $360^\circ$ circle an arc covers. To find arc length, take that same fraction of the circumference, using Desmos for the multiplication. Don't confuse the arc's angle in degrees with its length, and don't calculate a radius when the circumference is already given.

Hints

Hint 1
A central angle starts at the center of a circle. A full turn is $360^\circ$ , so what fraction of a turn does $120^\circ$ make?
Hint 2
An arc length is a distance along the circle, not an angle measured in degrees. The whole distance around is the circumference. What fraction of that distance does the arc take up?

Step-by-step

Take a fraction of the circumference
Step 1
Find the arc's share of the circle
A central angle has its tip at the circle's center. A full turn is $360^\circ$ , so the given $120^\circ$ angle covers $\frac{120}{360}=\frac13$ of the circle. Its intercepted minor arc, the shorter path between the angle's sides, covers that same fraction.
Step 2
Apply that share to the circumference
The circumference is the whole distance around the circle, and here it's $30$ . Arc length is the angle's fraction of that whole distance, so $\text{arc length}=\frac13(30)$ . Don't use $120$ as a length: it measures the angle in degrees.
Step 3
Find the arc length
Type $\frac13\cdot30$ in Desmos. It shows $10$ , so the minor arc is $10$ units long. Choice C.

Question 628·Easy·Math

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