math.. about sectors

2009-11-07 9:13 pm

Given a circular sector, is there always a different sector that has the same area and the same perimeter? Explain your answer.

My answer was that if the 2 variables (radius and angle theta) are equivalent in both sectors, then the area and perimeter for both will be the same. But how can I explain it in a more detailed way?

回答 (1)

用戶9112

2009-11-07 10:00 pm

✔ 最佳答案

Let the 2 sectors are having radii and angles r, θ and R,φ where the angles are in radians
Area = πr^2(θ/2π) = θr^2/2 = φR^2/2 => θ = φ(R/r)^2 ... (1)
Perimeter = rθ + 2r = Rφ + 2R
Use (1), r[φ(R/r)^2] + 2r - Rφ - 2R = 0
2r^2 - R(φ + 2)r + φR^2= 0
(r - R)(2r - Rφ) = 0
r = R(rejected, since the same) or r = Rφ/2
=> θ = φ(2R/Rφ)^2 = 4/φ
Therefore for a given sector of radius R and angle φ, we can always find another sector of radius Rφ/2 and angle 4/φ that has the same area and perimeter. The exception is when φ = 2, in such case the two sectors are the same.

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