The area of a square (L^2) - the area of a circle inscribed in that square (pi(L/2)^2) = (Approximately) The double integral of half of the ratio of the height with respect to the base of an equilateral triangle with sides L ( double integral ((1/2)(.86602)dL). (note that the ratio is fixed because its an equilateral triangle).

Can anyone shine some light on this relationship? Both of these formulas can also approximate some prime numbers, most likely because prime numbers cannot have
square roots and the area that is found from this equation finds the triangle shaped figures at the corners of a square. If using for prime numbers I believe it hits
2 3 5 7 11 when rounding.

I was bored. Perhaps, its nothing but me running into some silly and obvious relationship that can be more simply expressed.