An arc length is a distance along a circle, while a central angle tells you how much of a full turn that arc spans. Find the whole circumference, then match the arc’s fraction of it to the angle’s fraction of . A one-unknown Desmos regression finds the angle. Using half the circumference as the whole makes the angle twice as large.
Hints
- Hint 1
The circumference is the distance all the way around a circle. Use the radius in to find that whole distance before comparing it with the given arc length.
- Hint 2
A central angle has its vertex at the circle’s center. Its arc takes up the same fraction of the circumference that the angle takes up of a full turn. What equation matches those fractions?
Step-by-step
Match the arc and angle fractions
Step 1Find the whole circumference
No Desmos needed yet: this is one multiplication. The circumference is the distance around the whole circle. The radius is centimeters, so use :
The whole circumference is centimeters.
- Step 2
Match each part to its whole
The arc is centimeters long, so its fraction of the circumference is . Let mean the central angle in degrees. The arc’s fraction of the circumference equals the angle’s fraction of a full turn, so
The radius isn’t the whole distance; comparing the arc with it would make the angle too large.
- Step 3
Solve for the angle in Desmos
Type . Replace with and with so Desmos finds the value that makes the fractions match. Under PARAMETERS, it shows . So the central angle measures . Choice B.