Show that the solid angle subtended by a sphere of radius $r$ at a distance $R$ from its centre is $4 \pi r^2 / R^2$. Hence, by reasoning directly from the geometry, show that for a spherical body at the centre of a spherical cavity, the proportion of the radiation emitted from the cavity wall that hits the sphere is $r^2 / R^2$, if the emission is isotropic. Derive the same result using the shape factor reciprocity relation.