Let's form a regular hexagon by rotating it around: a) a diagonal passing through the center, $\alpha$, and b) a midline, $\beta$. How do the resulting solids of revolution compare in terms of:

a) their volumes,

b) their surface areas?
Let the side of the hexagon be $a$.

$\alpha)$ The first solid of revolution is composed of a cylinder and two congruent cones.

The height of the cylinder is $a$, and its radius is $\frac{a}{2} \sqrt{3}$.

The height of the cone is $\frac{a}{2}$, and its radius is also $\frac{a}{2} \sqrt{3}$.

The volume of the resulting solid of revolution is therefore

$$
K_{1}=\left(\frac{a}{2} \sqrt{3}\right)^{2} \pi a+2\left(\frac{a}{2} \sqrt{3}\right)^{2} \pi \frac{a}{6}=\frac{3 a^{3} \pi}{4}+\frac{a^{3} \pi}{4}=a^{3} \pi
$$

The surface area of the resulting solid of revolution is composed of the lateral surface area of the cylinder and the lateral surface areas of the two cones:

$$
F_{1}=2 \frac{a \sqrt{3}}{2} \pi a+2 \frac{a \sqrt{3}}{2} \pi a=2 \sqrt{3} a^{2} \pi
$$

$\beta$) The second solid of revolution is composed of two congruent frustums of cones. The radii of the bases are $a$ and $\frac{a}{2}$, and the height of the frustum is $\frac{a}{2} \sqrt{3}$.

The volume is therefore

$$
K_{2}=2 \frac{\frac{a \sqrt{3}}{2} \pi}{3}\left(a^{2}+\frac{a^{2}}{2}+\frac{a^{2}}{4}\right)=\frac{a \sqrt{3}}{3} \pi \cdot \frac{7 a^{2}}{4}=\frac{7 \sqrt{3}}{12} a^{3} \pi
$$

The surface area is the sum of the lateral surfaces of the two frustums and the areas of the two smaller bases:

$$
F_{2}=2 \frac{2 a \pi+a \pi}{2} a+2\left(\frac{a}{2}\right)^{2} \pi=3 a^{2} \pi+\frac{a^{2} \pi}{2}=\frac{7}{2} a^{2} \pi
$$

Thus, a) the ratio of the volumes is

$$
K_{1}: K_{2}=a^{3} \pi: \frac{7 \sqrt{3}}{12} a^{3} \pi=\frac{12}{7 \sqrt{3}}=\frac{4 \sqrt{3}}{7}
$$

b) the ratio of the surface areas is

$$
F_{1}: F_{2}=2 \sqrt{3} a^{2} \pi: \frac{7}{2} a^{2} \pi=\frac{4 \sqrt{3}}{7}
$$

Therefore, the ratio of the volumes and the ratio of the surface areas are the same: $\frac{4 \sqrt{3}}{7} \sim 0.99$.

Klára Madarász (Szeged, Tömörkény Secondary School, 2nd grade)