Circumscribed sphere

In

Existence and optimality

When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.^[5]

In De solidorum elementis (circa 1630),

Related concepts

The circumscribed sphere is the three-dimensional analogue of the circumscribed circle. All

Other spheres defined for some but not all polyhedra include a

When the circumscribed sphere is the set of infinite limiting points of hyperbolic space, a polyhedron that it circumscribes is known as an ideal polyhedron.

Point on the circumscribed sphere

There are five convex

. All Platonic solids have circumscribed spheres. For an arbitrary point ${\displaystyle M}$ on the circumscribed sphere of each Platonic solid with number of the vertices ${\displaystyle n}$, if ${\displaystyle MA_{i}}$ are the distances to the vertices ${\displaystyle A_{i}}$, then ^[8]

${\displaystyle 4(MA_{1}^{2}+MA_{2}^{2}+...+MA_{n}^{2})^{2}=3n(MA_{1}^{4}+MA_{2}^{4}+...+MA_{n}^{4}).}$

References

External links

Wikimedia Commons has media related to Circumscribed spheres.

• Weisstein, Eric W. "Circumsphere". MathWorld.