The insphere of a solid is a sphere that is tangent to all faces of the solid. An insphere does not always exist, but when it does, its radius r is called the inradius and its center the incenter. The insphere is the 3-dimensional generalization of the incircle.

Platonic solids (whose duals are themselves Platonic solids) and Archimedean duals have inspheres that touch all their faces, but Archimedean solids do not. Note that the insphere is not necessarily tangent at the centroid of the faces of a dual polyhedron, but is rather only tangent at some point lying on the face.


The figures above depict the inspheres of the Platonic solids.

The insphere is implemented in the Wolfram Language as Insphere[pts], where pts is a list of points determining a simplex, or Insphere[poly], where poly is a Polygon (giving a two-dimensional incircle) or Polyhedron (giving a three-dimensional insphere) object.

See also

Archimedean Dual, Circumsphere, Dual Polyhedron, Incenter, Incircle, Inradius, Midsphere, Platonic Solid

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Cite this as:

Weisstein, Eric W. "Insphere." From MathWorld--A Wolfram Web Resource.

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