A square and an equilateral triangle have the same perimeter. What is the ratio of the area of the circle circumscribing the square to the area of the circle inscribed in the triangle?

• let x be side of square
  perimeter of square=4x=perimeter of triangle=3*side of triangle
  so side of eq. triangle=(4/3)*x
  diameter of circle circumscribing the square=sqrt(2)*x
  area of circle circumscribing the square=pi*(sqrt(2)*x)^2/4=(pi/2)*x^2 ----(1)
  to find radius of the circle inscribed in the triangle
  area of triangle=r*s=sqrt(3)/4 * (4x/3)^2
  now s=(4/3)*x+(4/3)*x+(4/3)*x/2=2x
  so sqrt(3)/4 * (4x/3)^2=r*2x gives
  r={2/3*(3^1/2)}*x
  area of the circle inscribed in the triangle=pi*[{2/3*(3^1/2)}*x]^2
  =pi*(4/27)*x^2 -------(2)
  so reqd ratio= eqn(1)/eqn(2)
  =[(pi/2)*x^2]/[pi*(4/27)*x^2]=27/8
  so reqd ratio=27:8
  
  
  
  
• 11 years agoHelpfull: Yes(16) No(5)
• say the perimeter of the triangle and the square is x then side of the square will be x/4 and triangle x/3.
  now the radius of the circle1 circumscribing the square will be the 1/2 of the diagonal.so radius will be=x/4*(2^1/2).
  radius of the circle2 inscribed in the triangle will be= x/6*(3^1/2)
  so the answer will be 27:8
• 11 years agoHelpfull: Yes(11) No(5)