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?  Choose the correct choice from the following options
A) 9:8  B) 15:8  C) 27:8  D) 8:9  E) 8:15
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   ANSWER:27:8
<result>C</result>
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?