We place a sphere into a hollow, equilateral cone, which touches the cone's lateral surface at the base circumference. What fraction of the sphere's surface area lies inside the cone, and what fraction lies outside?
Let the radius of the sphere be $r$, the height of the slice lying in the cone be $m$; the part of the sphere's surface lying in the cone (cup):

$$
f=2 r \pi m
$$

but

$$
m=r-r \sin 30^{\circ}=r-\frac{r}{2}=\frac{r}{2}
$$

and thus

$$
f=r^{2} \pi
$$

The surface area of the sphere is $4 r^{2} \pi$, so we see that a quarter of the sphere's surface lies in the cone, and three-quarters of it lies outside.

The problem was solved by: Dolowschiák M., Flütsch I., Freibauer E., Kohn B., Rehberger Z., Spitzer Ö., Weisz J.