Question 1114707
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The expression for the area A of an equilateral triangle via its side length "a" is


A = {{{a^2*(sqrt(3)/4)}}},  which gives


a^2 = {{{(12*4)/sqrt(3)}}} = {{{(12*4*sqrt(3))/3}}} = {{{4^2*sqrt(3)}}}  ====>   a = {{{4*root(4,3)}}}.


Then the perimeter of the triangle is  P = 3a = {{{12*root(4,3)}}}.


Finally, the radius of the inscribed circle is  r = {{{(2A)/P}}} = {{{(2*4*root(4,3))/(12*root(4,3))}}} = {{{8/12}}} = {{{2/3}}} centimeters.


And the area of the inscribed circle is  {{{pi*r^2}}} = {{{pi*(2/3)^2}}} = {{{4pi/9}}}  cm^2.
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