Triakis Octahedron Calculator

Calculations at a triakis octahedron, the dual body of the truncated cube. A triakis octahedron is a regular octahedron with matching regular triangular pyramids attached to its faces. It has eight vertices with three edges and six vertices with eight edges.
Enter one value and choose the number of decimal places. Then click Calculate.

	Edge length octahedron (a):			24 faces, 36 edges, 14 vertices Faces: isosceles triangles
	Edge length pyramid (b):
	Surface area (A):
	Volume (V):
	Midsphere radius (rm):
	Insphere radius (ri):
	Surface-to-volume ratio (A/V):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

The triakis octahedron is a Catalan solid. Lengths and radiuses have the same unit (e.g. meter), the area 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.

Since three edges meet at every vertex of the truncated hexahedron, the faces of the triakis octahedron are triangles. Furthermore, because two edges of equal length, along with a third edge of a different length, meet at these vertices of the truncated hexahedron, the triangles of the triakis octahedron are isosceles. That is, they possess two edges of equal length and a third edge of a different length. This property stems from the duality between the Catalan solids and the Archimedean solids. For further details, please refer to the description of the triakis tetrahedron, the first of the Catalan solids in the conventional ordering. The sequence of the Archimedean solids is typically determined by the number of their faces, and the sequence of the Catalan solids is derived from this, as each Archimedean solid corresponds to a specific Catalan solid. Due to this duality, the number of solids in these two categories is equal.

The triakis octahedron is one of the few Catalan solids that can occur in nature in a similar form. Crystals belonging to the cubic crystal class sometimes adopt such a shape. Diamonds, for instance, occasionally do so. A triakis octahedron has a relatively simple structure, yet it appears significantly more complex. Although it is a convex form, the pointed tips of its pyramids give it a star-like appearance. The triakis octahedron can be described as a convex star, in contrast to non-convex star polyhedra such as the stellated octahedron. This makes this shape of particular interest for certain designs.

Calculation of volume and other dimensions: triakis octahedron

Last updated on 05/06/2026. Author: Jürgen Kummer

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