Question

If a square and an equilateral triangle have equal perimeters, what is the ratio of the area of the triangle to the area of the square?

Answer 1

`S_Delta/S_square = (4sqrt(3))/9`

Explanation:

Considering their equal perimeter, a side of an equilateral triangle `t` equals to `4/3` of a side of a square `s` because
`Perimeter = 3*t =4*s`

Therefore,
`t = 4/3s`

The altitude of an equilateral triangle with a side `t` equals to `tsqrt(3)/2`.
The area of an equilateral triangle with a side `t` is
`S_Delta = 1/2*t*tsqrt(3)/2 =t^2sqrt(3)/4`

The area of a square with a side `s` is
`S_square = s^2`

Their ratio is:
`S_Delta/S_square = (t^2sqrt(3))/(4s^2)`

Substituting `t = 4/3s`, we get
`S_Delta/S_square = (16s^2sqrt(3))/(9*4s^2) = (4sqrt(3))/9`