Volume of the Bicylinder and Tricylinder

Two of my favourite solids are formed by intersecting cylinders that are mutually perpendicular. When two cylinders intersect, the solid common to both cylinders is called a bicylinder. When three cylinders intersect, the solid common to both cylinders is called a tricylinder. Both the bicylinder and tricylinder now belong to a larger set of shapes called Steinmetz solids.

I wrote several articles on my wiki about these solids and how to determine their respective volumes.

Volume of the bicylinder:

https://math-physics-problems.wikia.org/wiki/Volume_of_the_Bicylinder

Volume of the tricylinder:

https://math-physics-problems.wikia.org/wiki/Volume_of_the_Tricylinder

tricylinder.png

Curiously, the bicylinder was used to figure out the volume of a sphere in ancient China. It was conceived by Liu Hui (c. 225 – 295 AD) during the Three Kingdoms Era. The volume was found by Zu Geng during the 5th century AD, the son of Zu Chongzhi (429 – 500 AD).

Here’s my article on how Zu Geng probably determined the solid (without integral calculus).

https://math-physics-problems.wikia.org/wiki/Liu_Hui%27s_Puzzle?action=edit&section=1  

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