Question

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=?`

Answer 1

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`