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Difficulty: EasyCircle Geometry: Arc Length and Sector Area

A circular pizza with a radius of 8 inches8\text{ inches} is cut into slices. If one slice has a central angle of 45∘45^\circ, what is the area, in square inches, of this slice?

  1. A
    2π2\pi
  2. 8π8\piAnswer
  3. C
    16π16\pi
  4. D
    56π56\pi
  5. E
    64π64\pi

Answer

The area of the slice is 8π8\pi square inches.
The area of a sector is found by multiplying the circle's total area by the fraction of the circle that the sector represents. With a radius of 88 inches, the total area is π×82=64π\pi \times 8^2 = 64\pi square inches. Since the slice has a central angle of 45∘45^\circ, it represents 45360=18\frac{45}{360} = \frac{1}{8} of the entire pizza. Multiplying the total area by this fraction gives 18×64π=8π\frac{1}{8} \times 64\pi = 8\pi square inches.

Step-by-Step Solution

1
Calculate the total area of the circular pizza using the area formula A=πr2A = \pi r^2 with r=8r = 8.
The total area of the pizza is π×82=64π\pi \times 8^2 = 64\pi square inches.
To find the area of a sector, we first need to determine the area of the entire circle.
2
Find the fraction of the circle represented by the slice's central angle by dividing 45∘45^\circ by 360∘360^\circ.
The fraction is 45360=18\frac{45}{360} = \frac{1}{8}.
A full circle has 360∘360^\circ, so the ratio of the central angle to 360∘360^\circ gives the proportion of the circle's total area that the sector occupies.
3
Multiply the total area of the pizza by the fraction representing the slice.
The area of the slice is 18×64π=8π\frac{1}{8} \times 64\pi = 8\pi square inches.
Multiplying the total circle area by the sector's fraction yields the sector's area.

Key Concept

The area of a sector is proportional to its central angle and can be found using the formula A=θ360πr2A = \frac{\theta}{360} \pi r^2 when the angle is in degrees.
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Topics

Circle Geometry: Arc Length and Sector Area

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