In geometry, a triakis octahedron (or trigonal trisoctahedron^[1] or kisoctahedron^[2]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

Triakis octahedron
(Click here for rotating model)
Type	Catalan solid
Coxeter diagram
Conway notation	kO
Face type	V3.8.8 isosceles triangle
Faces	24
Edges	36
Vertices	14
Vertices by type	8{3}+6{8}
Symmetry group	Oh, B3, [4,3], (*432)
Rotation group	O, [4,3]+, (432)
Dihedral angle	147°21′00″ arccos(−3 + 8√2/17)
Properties	convex, face-transitive
Truncated cube (dual polyhedron)	Net

It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.

This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.

If its shorter edges have length 1, its surface area and volume are:

${\displaystyle {\begin{aligned}A&=3{\sqrt {7+4{\sqrt {2}}}}\\V&={\frac {3+2{\sqrt {2}}}{2}}\end{aligned}}}$