SOLUTION: area of an inscribed semi circle in an equilateral triangle

1.  Make a sketch. Draw an equilateral triangle; an inscribed semi-circle; and the radius from the center of the semi-circle 
    to the tangent point on the triangle side.

    Notice that this radius is the height in the right-angled triangle which has the triangle side as the hypotenuse.


2.  Let "a" be the side length of the equilateral triangle.

    Then its area is  = .


3.  From the other side, the area of the equilateral triangle is twice the area of the right-angled triangle  = ar.

    Thus you get the equation 

     = ar,

    which gives you  r = .