\(GL = \frac{a}{3\sqrt{2}}\)
\(OG = \frac{a}{3}\)
\(GF = \frac{a}{3}\)
Moment of inertia \(I = I_1 + I_2 + I_3\)
\(I_1 = \frac{Ma^2}{12} + M\left(\frac{a}{3}\right)^2\)
\(I_2 = \frac{Ma^2}{12} + M\left(\frac{a}{3}\right)^2\)
\(I_3 = \frac{M(\sqrt{2}a)^2}{12} + M\left(\frac{a}{3\sqrt{2}}\right)^2\)
\(I = \frac{Ma^2}{12} + \frac{Ma^2}{12} + \frac{Ma^2}{9}\) \(+ \frac{Ma^2}{9} + \frac{2Ma^2}{12} + \frac{Ma^2}{18}\)
\(I = \frac{4Ma^2}{12} + \frac{5Ma^2}{18}\)
\(I = \frac{12Ma^2 + 10Ma^2}{36}\)
\(I = \frac{22Ma^2}{36}\)