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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A jeweler is creating a circular gold pendant with a radius of 1818 millimeters. A section of the pendant is shaped as a circular sector and has an arc length of 15π15\pi millimeters. What is the area, in square millimeters, of this sector?

  1. A
    30π30\pi
  2. B
    108π108\pi
  3. 135π135\piAnswer
  4. D
    189π189\pi
  5. E
    270π270\pi

Answer

The area of the sector is 135π135\pi square millimeters.
The sector area is calculated by applying the proportion of the arc length to the full circumference: Sector Area=Arc Length2πr×πr2=12rs\text{Sector Area} = \frac{\text{Arc Length}}{2\pi r} \times \pi r^2 = \frac{1}{2} r s. Substituting r=18r = 18 mm and s=15πs = 15\pi mm gives 12(18)(15π)=135π\frac{1}{2} (18)(15\pi) = 135\pi square millimeters.

Step-by-Step Solution

1
Find the ratio of the central angle to the full circle using arc length
The circumference of the circle is C=2πr=2π(18)=36πC = 2\pi r = 2\pi (18) = 36\pi mm. The sector's arc length fraction is 15π36π=512\frac{15\pi}{36\pi} = \frac{5}{12}.
Arc length is proportional to the total circumference of the circle.
2
Calculate the total area of the circle
Total Area=πr2=π(18)2=324π\text{Total Area} = \pi r^2 = \pi (18)^2 = 324\pi square millimeters.
The total area formula for a circle of radius rr is πr2\pi r^2.
3
Multiply the total area by the sector fraction to find the sector area
\text{Sector Area} = \frac{5}{12} \times 324\pi = 135\pi$ square millimeters.
The area of a circular sector is the same fraction of the total area as its arc length is of the circumference.

Key Concept

Relationship between arc length, radius, and sector area in a circle
Estimated Time:1m 0s
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Topics

Circle Geometry: Arc Length and Sector Area

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