Steinmetz solid

The common region inside equal-radius circular cylinders whose axes meet orthogonally. Two cylinders give a bicylinder; three give a tricylinder.

The solid intersection of two equal-radius infinite circular cylinders whose axes meet at a right angle. For radius $a$, the cross section at height $z$ along the common perpendicular coordinate is a square of area $4(a^2-z^2)$, so its volume is $16a^3/3$.

The common intersection of three equal-radius circular cylinders with mutually perpendicular axes. The displayed inequalities describe a centered example of radius $a$. It is a Steinmetz solid; its surface area is given by the surface area of a tricylinder formula.

For three radius-$a$ circular cylinders with mutually perpendicular axes through one point, the boundary of their common intersection has three congruent cylindrical surface pieces. On $x^2+y^2=a^2$, the height restriction is $|z|\le a\min (|\sin \theta |,|\cos \theta |)$. Eight congruent sectors give surface area $8a^2(2-\sqrt{2})$ per piece. Pairwise intersections are curves of zero surface area, so the three pieces add.