Question 604·Easy·Math

0 / 2000

In a circle with center $O$ , the measure of central angle $\angle AOB$ is $120^{\circ}$ , and the radius of the circle is 3.

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

A central angle has its vertex at the circle’s center. Compare its measure with a full $360^\circ$ turn to find the fraction of the circle it covers. The arc is that same fraction of the circumference, the distance around the full circle. Using the whole circumference gives the length of the entire circle, not the arc.

Hints

Hint 1
A central angle starts at the circle’s center. Compare its degree measure with the $360^\circ$ in a full turn. What fraction of the circle does the angle cover?
Hint 2
The circumference is the distance around the full circle, given by $C=\pi(2r)$ . Find it using the radius, then take the fraction you found from the angle.

Step-by-step

Take a fraction of the circumference
Step 1
Find the angle’s share of the circle
Because $O$ is the center, $\angle AOB$ is a central angle: it opens onto the arc from $A$ to $B$ . Compare $120^\circ$ with a full $360^\circ$ turn: $\frac{120}{360}=\frac13$ . So the minor arc covers one-third of the circle.
Step 2
Find the whole circumference
The circumference $C$ is the distance around the full circle. Substitute the radius $r=3$ into $C=\pi(2r)$ :
$C=\pi(2(3)).$
Multiply $2$ by $3$ :
$C=6\pi.$
Step 3
Take one-third of that length
The arc length is one-third of the circumference, so
$s=\frac13(6\pi).$
Keep $\pi$ outside the calculator and type $\frac{120}{360}\cdot2\cdot3$ to find its coefficient. Desmos prints $2$ , so minor arc $\widehat{AB}$ has length $2\pi$ . Choice D.

Question 604·Easy·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this