5. A right regular equilateral triangular prism is given, whose base has an area of $6.25 \sqrt{3} \mathrm{~cm}^{2}$.

a) Calculate the surface area of the given prism.

b) Determine the ratio of the volumes of the circumscribed and inscribed cylinders of this prism. Is this ratio the same for every right regular equilateral triangular prism?
Solution. a) From $\frac{a^{2} \sqrt{3}}{4}=6.25 \sqrt{3}$ we get $a=5 \text{ cm}$.

b) The radius of the inscribed cylinder is $r=\frac{1}{3} h$, and the radius of the circumscribed cylinder is $R=\frac{2}{3} h=2 r$, where $h$ is the height of the base. The considered cylinders have the same heights, so

$$
V_{1}: V_{2}=R^{2} \pi H: r^{2} \pi H
$$

i.e.

$$
V_{1}: V_{2}=R^{2}: r^{2}=(2 r)^{2}: r^{2}=4: 1
$$

where $V_{1}$ is the volume of the circumscribed cylinder, and $V_{2}$ is the volume of the inscribed cylinder. Thus, for all such prisms, the ratio of the considered volumes is constant.