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Difficulty: HardCircle Geometry

A sector of a circle with center OO has an area of 54π54\pi. The perimeter of the sector is 36+6π36 + 6\pi. If the radius of the circle is an integer, what is the radius of the circle?

Answer: 18

Answer

18
The area of a sector is given by A=12rsA = \frac{1}{2}rs, where rr is the radius and ss is the arc length. Given A=54πA = 54\pi, we have 12rs=54π\frac{1}{2}rs = 54\pi, which simplifies to rs=108πrs = 108\pi, or s=108πrs = \frac{108\pi}{r}. The perimeter of a sector is P=2r+sP = 2r + s. Given P=36+6πP = 36 + 6\pi, we can substitute ss to get 2r+108πr=36+6π2r + \frac{108\pi}{r} = 36 + 6\pi. Multiplying the entire equation by rr and rearranging terms yields the quadratic equation 2r2−(36+6π)r+108π=02r^2 - (36 + 6\pi)r + 108\pi = 0. Factoring by grouping gives (2r−6π)(r−18)=0(2r - 6\pi)(r - 18) = 0. This yields two potential solutions: r=3πr = 3\pi and r=18r = 18. Since the radius is specified to be an integer, the correct answer is 1818.

Step-by-Step Solution

1
Relate sector area and perimeter to radius and arc length
Area = 12rs=54π\frac{1}{2}rs = 54\pi and Perimeter = 2r+s=36+6π2r + s = 36 + 6\pi, where rr is the radius and ss is the arc length of the sector.
The area of a sector with radius rr and arc length ss is given by 12rs\frac{1}{2}rs, and its perimeter consists of the two radii plus the arc length.
2
Express arc length in terms of radius
s=108πrs = \frac{108\pi}{r}
Isolating ss from the area equation allows for substitution into the perimeter equation.
3
Substitute and form a quadratic equation
2r+108πr=36+6π  ⟹  2r2−(36+6π)r+108π=02r + \frac{108\pi}{r} = 36 + 6\pi \implies 2r^2 - (36 + 6\pi)r + 108\pi = 0
Multiplying both sides of the equation by rr eliminates the fraction and forms a standard quadratic equation.
4
Factor the quadratic equation
(2r−6π)(r−18)=0(2r - 6\pi)(r - 18) = 0
Grouping the terms as (2r2−36r)−(6πr−108π)=0(2r^2 - 36r) - (6\pi r - 108\pi) = 0 allows us to factor out 2r(r−18)−6π(r−18)=02r(r - 18) - 6\pi(r - 18) = 0.
5
Identify the integer radius
r=18r = 18
The two solutions to the equation are r=3πr = 3\pi and r=18r = 18. Since the problem specifies that the radius is an integer, we select 1818.

Key Concept

Calculating sector area and perimeter using relationships between radius, arc length, and angle measures.
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Topics

Circle Geometry

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