Moment of Inertia about the centroid of an equilateral triangle of side ‘a’ will be 

Explanation:

Consider an isosceles triangle:

\(A = \frac{{bh}}{2};\bar x = \frac{b}{2};\bar y = \frac{h}{3}\)

\({I_{xx}} = \frac{{b{h^3}}}{{36}};{I_{yy}} = \frac{{h{b^3}}}{{48}};{I_{xy}} = 0\)

\({I_{zz}} = {I_{xx}} + {I_{yy}} = \frac{{bh}}{{144}}\left( {4{h^2} + 3{b^2}} \right)\)

For an equilateral triangle:

\(h = \frac{{\sqrt 3 b}}{2}\)  and b = a

\({I_{xx}} = \frac{{b{h^3}}}{{36}} = {{a \over 36 } \times(\sqrt3 \times a/2 )}^3 = {{\sqrt 3} \times a^4 \over 96}\)