An 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? A. 3*sqrt(3) : 2 : sqrt(pi) B. 3^(3/4) : 2 : sqrt(pi) C. 3^(3/4) : sqrt(pi) : 2 D. sqrt(3) : 2 : sqrt(pi)

GATE 2022 · General Aptitude · Triangles · medium

Answer: P_triangle : P_square : P_circle = 3^(3/4) : 2 : sqrt(pi)

1. Find side a of the equilateral triangle: (sqrt(3)/4)*a^2 = A => a^2 = 4A/sqrt(3) = 4A*3^(-1/2) => a = 2*A^(1/2)*3^(-1/4)
2. Find side s of the square: s^2 = A => s = A^(1/2), so P_square = 4*A^(1/2)
3. Find radius r of the circle: pi*r^2 = A => r = (A/pi)^(1/2), so P_circle = 2*pi*r = 2*pi*(A/pi)^(1/2) = 2*(pi*A)^(1/2) = 2*A^(1/2)*sqrt(pi)
4. Form the ratio and cancel sqrt(A): = 6*A^(1/2)*3^(-1/4) : 4*A^(1/2) : 2*sqrt(pi)*A^(1/2). Divide each by 2*A^(1/2): = 3*3^(-1/4) : 2 : sqrt(pi) = 3^(1-1/4) : 2 : sqrt(pi) = 3^(3/4) : 2 : sqrt(pi)