# Extensive Definition

In geometry, the inscribed sphere
or insphere of a convex
polyhedron is a
sphere that is contained
within the polyhedron and tangent to each of the
polyhedron's faces. It is the largest sphere that is contained
wholly within the polyhedron, and is dual to the dual
polyhedron's circumsphere.

All regular
polyhedra have inscribed spheres, but some irregular polyhedra
do not have all facets tangent to a common sphere, although it is
still possible to define the largest contained sphere for such
shapes. For such cases, the notion of an insphere does not seem to
have been properly defined and various interpretations of an
insphere are to be found:

- The sphere tangent to all faces (if one exists).
- The sphere tangent to all face planes (if one exists).
- The sphere tangent to a given set of faces (if one exists).
- The largest sphere that can fit inside the polyhedron.

Often these spheres coincide, leading to
confusion as to exactly what properties define the insphere for
polyhedra where they do not coincide.

For example the regular
small stellated dodecahedron has a sphere tangent to all faces,
while a larger sphere can still be fitted inside the polyhedron.
Which is the insphere? Important authorities such as Coxeter or
Cundy & Rollett are clear enough that the face-tangent sphere
is the insphere. Again, such authorities agree that the Archimedean
polyhedra (having regular faces and equivalent vertices) have
no inspheres while the Archimedean dual or Catalan
polyhedra do have inspheres. But many authors fail to respect such
distinctions and assume other definitions for the 'inspheres' of
their polyhedra.

The radius of the sphere inscribed in a
polyhedron P is called the inradius of P.

## References

- Coxeter, H.S.M. Regular polytopes 3rd Edn. Dover (1973).
- Cundy, H.M. and Rollett, A.P. Mathematical Models, 2nd Edn. OUP (1961).

## See also

## External links

insphere in German: Inkugel

insphere in Esperanto: Enskribita sfero

insphere in Polish: Kula
wpisana