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Difficulty: MediumCircles

A circular steel plate is cut into 66 equal sectors. If the total area of the original plate was 144π cm2144\pi \text{ cm}^2, what is the arc length of the outer edge of one sector?

  1. 4π cm4\pi \text{ cm}Answer
  2. B
    2π cm2\pi \text{ cm}
  3. C
    24π cm24\pi \text{ cm}
  4. D
    12π cm12\pi \text{ cm}

Answer

The arc length of one sector is 4π cm4\pi \text{ cm}.
The correct answer is found by first using the total area (144π144\pi) to find the radius (r=12r = 12). Next, the full circumference is calculated using C=2πrC = 2\pi r, resulting in 24π24\pi. Finally, because the plate is divided into 6 equal sectors, the full circumference is divided by 6, giving an arc length of 4π cm4\pi \text{ cm}.

Step-by-Step Solution

1
Find the radius of the original circular plate.
Using the area formula A=πr2A = \pi r^2, substitute the given area: 144π=πr2144\pi = \pi r^2. Dividing by π\pi gives r2=144r^2 = 144, so the radius r=12 cmr = 12 \text{ cm}.
The radius is needed to calculate the total circumference of the circle.
2
Calculate the full circumference of the plate.
Using the circumference formula C=2πrC = 2\pi r, substitute r=12r = 12 to get C=2π(12)=24π cmC = 2\pi(12) = 24\pi \text{ cm}.
The arc length of a sector is a fraction of the circle's total circumference.
3
Determine the arc length of one of the 6 equal sectors.
Divide the full circumference by 6: 24π6=4π cm\frac{24\pi}{6} = 4\pi \text{ cm}.
Since the plate is cut into 6 equal sectors, the outer edge of one sector is exactly one-sixth of the total circumference.

Key Concept

The relationship between a circle's area, its circumference, and the proportional arc length of its sectors.
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Topics

Circles

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