In 3D, consider a large sphere (radius $d$) with a smaller sphere (radius $r$) centered on the large sphere's surface. It's surprising that the large sphere's surface covered by the small sphere has area $\pi r^2$, exactly, regardless of $d$.
In 2D, there is no simple relationship like this. The large circle's covered arc length is $2d\cdot\sin^{-1}{(r/d\cdot\sqrt{(1-r^2/4/d^2)})}$.
What about 4D? To reiterate, I'd like to know the "3D surface area" covered on a large sphere by a smaller sphere (centered on the large sphere's surface). If there is no simple analytic form, I'd like a first-order correction to the large $d$ limit of $4\pi r^3/3$ (of the large sphere's total $2\pi^2d^3$ surface area).