Sphere circumscribed about a prism

A sphere is said to be circumscribed about a prism if the sphere touches all the vertices (or sometimes all the faces) of the prism. For a right prism, this usually means the sphere passes through all the prism's vertices. Not every prism can have a circumscribed sphere; it depends on the shape of the base and the height.

Important properties

• For a sphere to be circumscribed about a prism, all the vertices of the prism must lie on the surface of the sphere.

• A necessary condition is that the base of the prism must be able to have a circumscribed circle (i.e., the base is a cyclic polygon).

• For a right prism with a regular polygon as its base, a circumscribed sphere exists if the height equals the diameter of the circumscribed circle of the base.

• The center of the circumscribed sphere is equidistant from all the prism's vertices.