Removing a central cylindrical hole from a sphere leaves a solid whose volume depends only on its remaining height . With , sphere radius and hole radius , the rim gives . Every horizontal slice is an annulus of area , so integrating from to gives . This slice equality explains why changing while retaining leaves the volume unchanged.
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 , the cross section at height along the common perpendicular coordinate is a square of area , so its volume is .
The common intersection of three equal-radius circular cylinders with mutually perpendicular axes. The displayed inequalities describe a centered example of radius . It is a Steinmetz solid; its surface area is given by the surface area of a tricylinder formula.
For three radius- circular cylinders with mutually perpendicular axes through one point, the boundary of their common intersection has three congruent cylindrical surface pieces. On , the height restriction is . Eight congruent sectors give surface area per piece. Pairwise intersections are curves of zero surface area, so the three pieces add.
