Here is the solution from Christiane Eaves, Alicia Maltby, Kathy Lam, Rachael Evans & Fiona Conroy (Y10) The Mount School,York: "

The total surface area of the cube is $6r^2.$ The area shaded on one face is $\frac{\pi r^2}{4}$ so the total shaded area is $\frac{3\pi r^2}{4}.$

The fraction of the total surface area shaded is thus $\frac{3\pi r^2)}{4}$ divided by $6r^2$ which is:

${\displaystyle \frac{3\pi r^2}{4}\times \frac{1}{6r^2}=\frac{\pi }{8}}$