Question 652·Easy·Math

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In the circle shown, radii $\overline{OA}$ and $\overline{OB}$ form a central angle of $60^\circ$ . The circle has a circumference of 24.

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

An arc length is the distance along part of a circle. A central angle tells you what fraction of a full $360^\circ$ turn that arc covers, so take the same fraction of the circumference. The key trap is treating the angle’s degree measure as a length, or treating a small angle as a quarter turn without checking its size.

Hints

Hint 1
A central angle has its corner at the circle’s center. The arc between its radii takes the same fraction of the circle as the angle takes of a full $360^\circ$ turn. What is that fraction?
Hint 2
The circumference is the distance all the way around the circle. The arc is part of that distance, so multiply the given circumference by the fraction you found.

Step-by-step

Take a fraction of the circumference
Step 1
Find the arc’s share of the circle
$O$ is the center, so the $60^\circ$ angle between the radii is a central angle. A full turn is $360^\circ$ , so the minor arc between $A$ and $B$ takes up:
$\frac{60}{360}=\frac16$
Step 2
Match that share to the circumference
The circumference is the whole distance around the circle, and the problem gives it as $24$ . The arc takes the same fraction of the circumference as its central angle takes of a full turn. So its length, $L$ , is:
$L=\frac16(24)$
Step 3
Calculate the arc length
Type $\frac16\cdot24$ in Desmos. It shows $4$ , so the length of minor arc $\widehat{AB}$ is $4$ . Choice B.

Question 652·Easy·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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