8. Carlo Emilio Gadda writes in one of the stories from L'Adalgisa—Milanese Sketches: ``The service rooms, the bathroom, the corridors, the antechamber and one of the two toilets were paved with small red tiles: hexagonal [. . .]. The apothem of those tiles measured 5.196 centimeters: whereas the radius of the circumscribed circle reached 60 millimeters''. Express the exact relationship between the radius of the circumscribed circle and the apothem (that is, the radius of the inscribed circle) for a regular hexagon. Verify the result obtained in light of the measurements indicated by the writer. Explain why, using regular hexagonal tiles all congruent to each other, it is possible to tile a plane. With which other regular polygons, congruent to each other, is it possible to tile a plane? Justify your answer.
\footnotetext{Maximum duration of the test: 6 hours. The use of scientific or graphical calculators is permitted provided they are not equipped with symbolic algebraic processing capability and do not have Internet connectivity. The use of a bilingual dictionary (Italian–language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the distribution of the test. }