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Question

Difficulty: EasyCircle Geometry

A circle has a radius of 1010. A shaded sector of the circle has an area of 10π10\pi. What is the measure, in degrees, of the central angle of the shaded sector?

Answer: 36 degrees

Answer

The measure of the central angle of the shaded sector is 36 degrees.
The total area of the circle is A=π(10)2=100πA = \pi (10)^2 = 100\pi. The shaded sector's area is 10π10\pi, which represents 10π100π=110\frac{10\pi}{100\pi} = \frac{1}{10} of the total area of the circle. Since a circle consists of 360∘360^\circ, the central angle of the sector is 110×360∘=36∘\frac{1}{10} \times 360^\circ = 36^\circ.

Step-by-Step Solution

1
Calculate the total area of the circle.
Total area = 100π100\pi
The area of a circle is calculated using the formula A=πr2A = \pi r^2 where the radius r=10r = 10.
2
Determine the proportion of the circle covered by the shaded sector.
Proportion = 110\frac{1}{10}
Dividing the sector's area of 10π10\pi by the total area of 100π100\pi gives the fraction of the circle represented by the sector.
3
Find the central angle in degrees.
Central angle = 36
Since a full circle has a central angle of 360∘360^\circ, multiplying the proportion 110\frac{1}{10} by 360∘360^\circ yields the central angle of the sector.

Key Concept

The ratio of the area of a sector to the total area of a circle is equal to the ratio of the sector's central angle measure to the total degree measure of a circle (360∘360^\circ).
Estimated Time:45s
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Topics

Circle Geometry

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