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In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing. 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. All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have all vertices lying on a common sphere, although it is still possible to define the smallest containing sphere for such shapes.
The radius of sphere circumscribed around a polyhedron is called the circumradius of </math>P</math> .
The volume of sphere circumscribed around a polyhedron is therefore:
The surface area of sphere circumscribed around a polyhedron is therefore:
The volume of the hemisphere circumscribed in a polyhedron is therefore:
The surface area of the hemisphere circumscribed in a polyhedron is therefore: