A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the square and B be the area of the circle cicumscribed about the triangle. Then #A/B=?#

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Answer 1 · 2018-07-29T13:48:48

Let the common perimeter be #c#
So each side of the square #c/4#
And the circumradius of the square #(r_s)=1/2sqrt2xxc/4=c/(4sqrt2)#

So the area of the circle circumscribed about the square will be

#A=pir_s^2=pic^2/32#

Each side of the triangle will be #c/3#

Its height #h=sqrt3/2xxc/3#

The circumradius of the triangle #(r_t)=2/3xxsqrt3/2xxc/3=c/(3sqrt3)#

So area of the circle cicumscribed about the triangle.

#B=pir_t^2=pic^2/27#

Hence the ratio #A/B=27/32#