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Difficulty: EasyCircles, Arc Lengths, and Sector Areas

A sector of a circle with a radius of 1212 units has an area of 24π24\pi square units. What is the measure, in degrees, of the central angle of the sector?

Answer: 60 degrees

Answer

The measure of the central angle of the sector is 60∘60^\circ.
The area of the full circle is πr2=π(12)2=144π\pi r^2 = \pi (12)^2 = 144\pi. The sector area represents a fraction of the total area, specifically 24π144π=16\frac{24\pi}{144\pi} = \frac{1}{6}. Multiplying this fraction by the full circle's central angle of 360∘360^\circ gives 16×360∘=60∘\frac{1}{6} \times 360^\circ = 60^\circ.

Step-by-Step Solution

1
Find the total area of the circle
The total area of the circle is π×122=144π\pi \times 12^2 = 144\pi.
The area of a full circle with radius rr is given by A=πr2A = \pi r^2.
2
Relate the sector area to the total circle area
The fraction of the circle represented by the sector is 24π144π=16\frac{24\pi}{144\pi} = \frac{1}{6}.
The area of a sector is proportional to the fraction of the total central angle (360∘360^\circ) it subtends.
3
Calculate the central angle in degrees
\theta = \frac{1}{6} \times 360^\circ = 60^\circ.
Multiply the fraction of the circle by 360∘360^\circ to get the central angle measure.

Key Concept

Relationship between central angle measure, total circle area, and sector area
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Topics

Circles, Arc Lengths, and Sector Areas

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