A semi-circular lamina of mass m has radius r and centre C. Its moment of inertia about an axis through its centre of mass and perpendicular to its plane is $(\frac{1}{2}-\frac{16}{k{\pi }^2})m{r}^2$ , then value of k is

For the semi-circular lamina of mass m, the moment of inertia about an axis through C is $I_C=\frac{1}{2}m{r}^2$ . 

Let $I_{CM}$ =moment of inertia about an axis through its centre of mass $I_C=I_{CM}+m{d}^2\Rightarrow I_{CM}=I_C-m{d}^2=\frac{1}{2}m{r}^2-m{\left(\frac{4r}{3k}\right)}^2=(\frac{1}{2}-\frac{16}{9k^2})m{r}^2$