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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A circular tabletop has a radius of 3030 inches. A wooden section shaped as a circular sector covers a portion of the tabletop defined by a central angle of 108∘108^\circ. What is the area, in square inches, of this wooden sector section?

  1. A
    9π9\pi
  2. B
    18π18\pi
  3. 270π270\piAnswer
  4. D
    540π540\pi
  5. E
    1080π1080\pi

Answer

The area of the wooden sector section is 270π270\pi square inches.
The area of a sector of a circle with radius rr and central angle θ\theta (in degrees) is given by the formula A=θ360∘⋅πr2A = \frac{\theta}{360^\circ} \cdot \pi r^2. Substituting r=30r = 30 inches and θ=108∘\theta = 108^\circ yields A=108360⋅π(30)2=310⋅900π=270πA = \frac{108}{360} \cdot \pi (30)^2 = \frac{3}{10} \cdot 900\pi = 270\pi square inches.

Step-by-Step Solution

1
Calculate the total area of the circular tabletop.
Total Area = πr2=π(30)2=900π\pi r^2 = \pi (30)^2 = 900\pi square inches.
The area of a full circle with radius rr is given by A=πr2A = \pi r^2.
2
Determine the fraction of the circle represented by the central angle.
Fraction = 108∘360∘=310\frac{108^\circ}{360^\circ} = \frac{3}{10}.
A full circle measures 360∘360^\circ, so the central angle forms a fraction θ360∘\frac{\theta}{360^\circ} of the total circle.
3
Multiply the fraction by the total area of the circle to find the sector area.
Sector Area = 310⋅900π=270π\frac{3}{10} \cdot 900\pi = 270\pi square inches.
The sector area is proportional to the fraction of the central angle relative to the full circle.

Key Concept

Sector Area Formula
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Topics

Circle Geometry: Arc Length and Sector Area

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