Steinmetz Solids Calculator: Bicylinder and Tricylinder

Name: Steinmetz Solids: Bicylinder and Tricylinder - Geometry Calculator
Author: Jürgen Kummer

Calculations with Steimetz solids. Those solids, names after Charles Steinmetz, are generated by the intersection of two or three cylinders of equal radius at right angles. The intersection of two cylinders is the bicylinder, that of three is the tricylinder.
Choose the type of Steimetz solid, enter one value and choose the number of decimal places. Then click Calculate.

The radius has a one-dimensional unit (e.g. meter), the surface has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

Charles Proteus Steinmetz, born Carl August Rudolph Steinmetz, was a German-American electrical engineer who fled Germany for the USA in 1889 due to political persecution. In addition to his contributions to electrical engineering, he calculated the intersections of two and three solid cylinders. Solid cylinder refers to the entire object, including its interior. The term cylinder here refers specifically to right circular cylinders, not to generalized cylinders. While the bicylinder and its volume were already known to Archimedes 2200 years earlier, the tricylinder was not. Steinmetz brought both shapes to prominence.
Both the bicylinder and the tricylinder exhibit point symmetry about their center. The bicylinder has five planes of symmetry, whereas the tricylinder has nine. The tricylinder also possesses a significantly higher degree of rotational symmetry than the bicylinder. The bicylinder exhibits square symmetry, while the tricylinder exhibits cubic symmetry. The bicylinder has four curved faces, two edges, and no vertices, or four edges and two vertices, if the apices are regarded as vertices. The tricylinder has 8 apices, 14 vertices, 24 edges, and 12 faces. The apices of the tricylinder are the vertices where three edges meet, whereas four edges meet at the other vertices. The surface decomposition of the tricylinder formed by the intersection curves has the same vertex and edge structure as a rhombic dodecahedron.

Steinmetz solids are formed as cavities where pipelines and boreholes intersect. In architecture, the groin vault resembles half a bicylinder. In nature, fused plant stems can produce similar shapes.

Calculation of volume and other dimensions: steinmetz solids: bicylinder and tricylinder

Last updated on 06/09/2026. Author: Jürgen Kummer

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