Flux passing through shaded surface of sphere when a point charge q is placed at the centre is (radius of the sphere is R)

α =6O°
Solid angle subtended by BCD,
$\mathrm{\Omega }=2\mathrm{\pi }(1-\mathrm{cos\alpha })=\mathrm{\pi }$
Solid angle subtended by ABDE,
${\mathrm{\Omega }}_{\left(\mathrm{ABCDE}\right)}-{\mathrm{\Omega }}_{\left(\mathrm{BCD}\right)}=2\mathrm{\pi }-\mathrm{\pi }=\mathrm{\pi }$

Hence, flux through ABDE: $\mathrm{\varphi }=\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}\frac{\mathrm{\pi }}{4\mathrm{\pi }}=\frac{\mathrm{q}}{4{\mathrm{\epsilon }}_0}$