$\mathrm{\%}\mathrm{packing}\mathrm{efficiency}=\frac{\mathrm{Vol}.\mathrm{occupied}\mathrm{by}\mathrm{atom}}{\mathrm{Vol}.\mathrm{of}\mathrm{unit}\mathrm{cell}}\times 100\frac{\frac{4}{3}{\mathrm{\pi r}}^3}{{\mathrm{a}}^3}\times 100$

Let radius fo corner atom is r and radius of central atom is 2r

$\mathrm{So},\sqrt{3}\mathrm{a}=2\left(2\mathrm{r}\right)+2\mathrm{r}=6\mathrm{r}$

$\mathrm{a}=\frac{6\mathrm{r}}{\sqrt{3}}=2\sqrt{3}\mathrm{r}$

Now

$\mathrm{\%}\mathrm{P}.\mathrm{E}.=\frac{\frac{4}{3}{\mathrm{\pi r}}^3+\frac{4}{3}\mathrm{\pi }{\left(2\mathrm{r}\right)}^3}{{\left(2\sqrt{3}\mathrm{r}\right)}^3}\times 100$

$=\frac{\frac{4}{3}\mathrm{\pi }({\mathrm{r}}^3+8{\mathrm{r}}^3)}{8\times 3\sqrt{3}{\mathrm{r}}^3}\times 100$

$=\frac{4\mathrm{\pi }\times 9{\mathrm{r}}^3}{3\times 8\times 3\sqrt{3}{\mathrm{r}}^3}\times 100=90.6\mathrm{\%}\approx 90\mathrm{\%}$