Question 692·Medium·Math

0 / 2000

In a circle with center $O$ , a central angle measures $\dfrac{2\pi}{3}$ radians and intercepts an arc whose length is $12\pi$ centimeters. What is the radius of the circle, in centimeters?

Your answer

An arc length is the distance along a circle, not the size of its central angle. If the angle is in radians, the formula $s=r\theta$ connects that distance to the radius. Put the given arc length and angle into the formula, then divide by the angle. Don’t treat the arc as the whole circumference.

Hints

Hint 1
Radians measure an angle, while centimeters measure the arc’s distance. For an angle in radians, $s=r\theta$ , where $s$ is arc length and $r$ is radius. Which given values belong in that equation?
Hint 2
The radius is multiplied by the angle in the arc-length formula. Divide the arc length by the angle to undo that multiplication. Keep $\pi$ in both values as you divide; what happens to it?

Step-by-step

Use the radian arc-length formula
Step 1
Turn the angle and arc length into an equation
A central angle has its vertex at the circle’s center. Because this angle is in radians, arc length equals radius times the angle: $s=r\theta$ . Here $s$ is arc length and $\theta$ is the angle, so the given values make the equation:
$12\pi=r\left(\frac{2\pi}{3}\right).$
Step 2
Divide to find the radius
Divide both sides by the angle $\frac{2\pi}{3}$ , which multiplies the radius: $r=\frac{12\pi}{2\pi/3}$ .Type $\frac{12\pi}{2\pi/3}$ in Desmos. It displays $18$ ; the $\pi$ factors cancel. So the radius is $18$ centimeters. Grid in 18.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreFind arc length, sector area, and radian measure

Question 692·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this