Project ForgeAn equilateral triangle, a square and a circle have equal areas. What is the ratio of the perimeters of the equilateral triangle to square to circle?
- A3*sqrt(3) : 2 : sqrt(pi)
- sqrt(3*sqrt(3)) : 2 : sqrt(pi)
- Csqrt(3*sqrt(3)) : 4 : 2*sqrt(pi)
- Dsqrt(3*sqrt(3)) : 2 : 2*sqrt(pi)
Explanation
For equal area A: triangle side = 2*sqrt(A)/3^(1/4), so perimeter = 6*sqrt(A)/3^(1/4) = 1.5*3^(-1/4) x 4*sqrt(A). Square side = sqrt(A), perimeter = 4*sqrt(A). Circle radius = sqrt(A/pi), perimeter = 2*sqrt(pi*A). Working out the triangle:square ratio gives exactly sqrt(3*sqrt(3)):2 (since 3^0.75/2 = 1.5*3^(-0.25)), and the square:circle ratio gives exactly 2:sqrt(pi) (since 4:2*sqrt(pi) simplifies to 2:sqrt(pi)) — matching sqrt(3*sqrt(3)):2:sqrt(pi) exactly.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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