Moment of inertia of rectangular lamina about its diagonal is

Given that a rectangular lamina of length l and breath b it is divided into two triangular plates about diagonal each have mass $=\frac{\text{M}}{\text{2}}$

We can have height of each triangular portion about diagonal = r

$\frac{\text{r}}{\text{b}}=\frac{l}{\sqrt{l^2+{\text{b}}^{\text{2}}}}$

$\text{r}=\frac{l\text{b}}{l^2+{\text{b}}^{\text{2}}}$

M.I of Lamina about diagonal = 2(M.I of the triangular plates as shown in fig)

$I=2\left(\frac{1}{6}\left(\frac{\text{M}}{\text{2}}\right){\text{r}}^{\text{2}}\right)$

$=\frac{1}{6}M{\left(\frac{l\text{b}}{l^2+{\text{b}}^{\text{2}}}\right)}^2$

$=\frac{1}{6}\frac{\text{M}{l}^2{\text{b}}^{\text{2}}}{l^2+{\text{b}}^{\text{2}}}$