Question 784·Medium·Math

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In circle $O$ , the radius is $10\text{ cm}$ . If the measure of minor arc $AB$ is $72^\circ$ , what is the length, in centimeters, of arc $AB$ ?

An arc length is the distance along part of a circle’s edge. An arc’s angle in degrees tells you what fraction of the full $360^\circ$ circle that part covers. Take the same fraction of the circumference, the distance around the whole circle. In $2\pi r$ , use the center-to-edge radius, not the distance all the way across.

Hints

Hint 1
An arc’s degree measure tells you its share of a full $360^\circ$ turn; it doesn’t tell you its length in centimeters. What fraction of the circle does $72^\circ$ cover?
Hint 2
The circumference is the distance around the whole circle: $C=2\pi r$ . Use the radius to find that whole distance, then take the fraction you found to get the arc’s length.

Step-by-step

Take a fraction of the circumference
Step 1
Find the arc’s fraction of a circle
No Desmos needed. The angle gives a fraction that reduces quickly. An arc’s degree measure is an angle, not a length in centimeters. Divide the given $72^\circ$ by a full $360^\circ$ turn:
$\frac{72}{360}=\frac15.$
Step 2
Find the whole circumference
The radius runs from the center to the circle, so the given $10$ cm goes into $C=2\pi r$ :
$C=2\pi(10).$
Multiply $2$ by $10$ :
$C=20\pi.$
Step 3
Find the length of arc AB
Let $s$ be the length of arc $AB$ . The arc covers the same fraction of the circumference as its angle covers of a full turn. Multiply the one-fifth from the first step by the circumference:
$s=\frac15(20\pi).$
Divide $20$ by $5$ :
$s=4\pi.$
So arc $AB$ is $4\pi$ centimeters long. Choice D.

Question 784·Medium·Math

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