Tricylinder ($x^2+y^2\le a^2,y^2+z^2\le a^2,z^2+x^2\le a^2$)

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.