Question 1063362
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area of an inscribed semi circle in an equilateral triangle
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<pre>
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 {{{(1/2)*a*(a*sqrt(3)/2)}}} = {{{(a^2*sqrt(3))/4}}}.


3.  From the other side, the area of the equilateral triangle is twice the area of the right-angled triangle {{{2*((a*r)/2))}}} = ar.

    Thus you get the equation 

    {{{(a^2*sqrt(3))/4}}} = ar,

    which gives you  r = {{{(a*sqrt(3))/4}}}.
</pre>

Solved.