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Difficulty: MediumCircles, Arc Lengths, and Sector Areas

A sector of a circle has an area of 15π15\pi square units and a perimeter of 10+6π10 + 6\pi units. What is the measure of the central angle of the sector, in degrees?

  1. A
    108∘108^\circ
  2. B
    135∘135^\circ
  3. C
    144∘144^\circ
  4. 216∘216^\circAnswer
  5. E
    270∘270^\circ

Answer

The correct answer is 216∘216^\circ, corresponding to the option stating 216∘216^\circ.
The perimeter of a sector is defined as 2r+s=10+6π2r + s = 10 + 6\pi, which yields a radius of r=5r = 5 and an arc length s=6πs = 6\pi. Checking with the sector area formula 12rs=12(5)(6π)=15π\frac{1}{2}rs = \frac{1}{2}(5)(6\pi) = 15\pi confirms these measurements. The total area of the circle is π(5)2=25π\pi (5)^2 = 25\pi. The sector thus constitutes 15π25π=35\frac{15\pi}{25\pi} = \frac{3}{5} of the circle. Multiplying this fraction by 360∘360^\circ gives a central angle of 216∘216^\circ.

Step-by-Step Solution

1
Express the sector perimeter formula and solve for radius and arc length.
Radius r=5r = 5 and arc length s=6πs = 6\pi.
The perimeter of a sector equals two radii plus its arc length: Perimeter=2r+s=10+6π\text{Perimeter} = 2r + s = 10 + 6\pi. Equating standard and π\pi-termed components gives 2r=10  ⟹  r=52r = 10 \implies r = 5 and arc length s=6πs = 6\pi.
2
Verify consistency using the sector area formula.
Area=15π\text{Area} = 15\pi, matching the given information.
The area of a sector can also be calculated as 12rs=12(5)(6π)=15π\frac{1}{2} r s = \frac{1}{2}(5)(6\pi) = 15\pi.
3
Find the total area of the circle and the fraction of the circle occupied by the sector.
Total area =25π= 25\pi, area fraction =35= \frac{3}{5}.
The full circle area is πr2=π(52)=25π\pi r^2 = \pi (5^2) = 25\pi. The sector represents 15π25π=35\frac{15\pi}{25\pi} = \frac{3}{5} of the entire circle.
4
Calculate the central angle θ\theta in degrees.
θ=216∘\theta = 216^\circ.
Multiply the fraction by 360∘360^\circ: θ=35×360∘=216∘\theta = \frac{3}{5} \times 360^\circ = 216^\circ.

Key Concept

Perimeter, Arc Length, and Area of a Circle Sector
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Topics

Circles, Arc Lengths, and Sector Areas

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