Question 640·Easy·Math

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In a circle with center $O$ , the central angle $\angle AOB$ measures $60^{\circ}$ . The circle has a circumference of 30.

What is the length of minor arc $\widehat{AB}$ ?

A central angle has its vertex at the circle’s center. Its share of a full $360^{\circ}$ turn tells you what share of the whole circumference belongs to the arc. Multiply that fraction by the circumference; Desmos can handle the arithmetic. Don’t confuse an arc’s measure in degrees with its physical length.

Hints

Hint 1
A central angle starts at the circle’s center, so it cuts out the same fraction of the circle as its angle is of a full $360^{\circ}$ turn. What fraction does $60^{\circ}$ represent?
Hint 2
An arc length is part of the circle’s circumference, the distance all the way around. Multiply the angle’s fraction by the given circumference, not by the angle’s degree measure.

Step-by-step

Take a fraction of the circumference
Step 1
Find the angle’s share of a full turn
A central angle has its vertex at the center $O$ . A full turn is $360^{\circ}$ , so the given $60^{\circ}$ angle cuts out one-sixth of the circle:
$\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}.$
Step 2
Turn that share into an arc length
An arc takes the same fraction of the circumference as its central angle takes of a full turn. Multiply by the given circumference of $30$ :
$\text{arc length}=\frac{60}{360}(30).$
The $60^{\circ}$ measures an angle, not a length.
Step 3
Calculate the length
Type $\frac{60}{360}\cdot30$ into Desmos. It displays $5$ , the length of minor arc $\widehat{AB}$ . Choice B.

Question 640·Easy·Math

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