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Difficulty: MediumCircle Geometry

A circle with center OO has a radius of 88. Points AA and BB lie on the circle such that the length of the minor arc ABAB is 5π5\pi. What is the area of the major sector AOBAOB?

  1. A
    20π20\pi
  2. B
    36π36\pi
  3. 44π44\piAnswer
  4. D
    11π11\pi

Answer

The area of the major sector is 44π44\pi.
The area of the major sector is found by scaling the total area of the circle, 64π64\pi, by the ratio of the major arc length to the circumference. The circumference is 2π(8)=16π2\pi(8) = 16\pi. Since the minor arc length is 5π5\pi, the major arc length is 16π−5π=11π16\pi - 5\pi = 11\pi. The ratio of the major arc to the circumference is 11π16π=1116\frac{11\pi}{16\pi} = \frac{11}{16}. Multiplying this ratio by the total area of the circle yields 1116×64π=44π\frac{11}{16} \times 64\pi = 44\pi.

Step-by-Step Solution

1
Calculate the circumference of the circle.
16π16\pi
The circumference formula is C=2πrC = 2\pi r. Given r=8r = 8, the circumference is 2π(8)=16π2\pi(8) = 16\pi.
2
Find the length of the major arc ABAB.
11π11\pi
The length of the major arc is the total circumference minus the length of the minor arc: 16π−5π=11π16\pi - 5\pi = 11\pi.
3
Calculate the total area of the circle.
64π64\pi
The area formula for a circle is A=πr2A = \pi r^2. Given r=8r = 8, the total area is π(82)=64π\pi(8^2) = 64\pi.
4
Determine the area of the major sector by scaling the total area.
44π44\pi
The major sector's area is proportional to the fraction of the circle represented by the major arc: 11π16π×64π=1116×64π=44π\frac{11\pi}{16\pi} \times 64\pi = \frac{11}{16} \times 64\pi = 44\pi.

Key Concept

Calculating sector area using arc length and total circle area relationships.
Estimated Time:1m 30s
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Topics

Circle Geometry

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