Archimedes' famous-theorem

TL;DR

Archimedes proved that a sphere's volume is two-thirds that of its circumscribing cylinder, likely using a clever proof involving the Pythagorean theorem rather than the traditional polygon rotation method.

Contribution

The paper suggests a shorter, more elegant proof of Archimedes' theorem that does not rely on π or polygon rotations, highlighting a novel approach.

Findings

Archimedes' proof may have used the Pythagorean theorem.

A shorter demonstration of the theorem is possible.

Traditional methods involve polygon rotations and π.

Abstract

In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures to obtain a set of trunks in a sphere. This set of trunks allowed him to determine the volume. In our opinion, Archimedes was so clever that he found the proof with shorter demonstration. Archimedes did not need to know $\pi$ to prove the result and the Pythagorean theorem was probably the key to the proof.