Question 1014340
So the radius of the circle is,
{{{2pi*R=50}}}
{{{R=25/pi}}}
The equilateral triangle inscribed in the circle is made up of 3 isosceles triangles that have 2 sides equal to the radius of the circle and the third side equal to the length of the side of the equilateral triangle. 
*[illustration dx9.JPG].
From trigonometry,
{{{cos(30)=(S/2)/R}}}
{{{S=2Rcos(30)}}}
For an equlateral triangle,
{{{P=3S}}}
{{{P=6Rcos(30)}}}
{{{P=6(25/pi)(sqrt(3)/2)}}}
{{{P=(75sqrt(3))/pi)}}}
{{{highlight(P=(75sqrt(3))/pi))}}}