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Question

Difficulty: EasyCircle Geometry

A circle has a circumference of 16π16\pi. An arc on this circle has a length of 2π2\pi. What is the measure, in degrees, of the central angle that subtends this arc?

  1. A
    22.522.5
  2. B
    4040
  3. 4545Answer
  4. D
    9090

Answer

45
The correct answer is 4545 because the ratio of the arc length to the total circumference is 2π16π=18\frac{2\pi}{16\pi} = \frac{1}{8}. Since a full circle measures 360∘360^\circ, the central angle that subtends this arc is 18×360∘=45∘\frac{1}{8} \times 360^\circ = 45^\circ.

Step-by-Step Solution

1
Find the ratio of the arc length to the total circumference of the circle.
The ratio is 2π16π=18\frac{2\pi}{16\pi} = \frac{1}{8}.
An arc length represents a fraction of the circle's total perimeter (circumference).
2
Multiply this fraction by the total degree measure of a circle (360∘360^\circ) to find the measure of the central angle.
18×360∘=45∘\frac{1}{8} \times 360^\circ = 45^\circ.
The measure of the central angle is proportional to the fraction of the circle's circumference that the arc covers.

Key Concept

The relationship between arc length, circumference, and central angle measure in a circle.
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Topics

Circle Geometry

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