Ingeometry, acircumscribed sphere of apolyhedron is asphere that contains the polyhedron and touches each of the polyhedron'svertices.^[1] The wordcircumsphere is sometimes used to mean the same thing, by analogy with the termcircumcircle.^[2] As in the case of two-dimensional circumscribed circles (circumcircles), theradius of a sphere circumscribed around a polyhedronP is called thecircumradius ofP,^[3] and the center point of this sphere is called thecircumcenter ofP.^[4]

When it exists, a circumscribed sphere need not be thesmallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of acube 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 theconvex hull of a subset of the vertices of the polyhedron.^[5]

InDe solidorum elementis (circa 1630),René Descartes observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: somebipyramids, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever asimple polyhedron has a circumscribed circle for each of its faces, it also has a circumscribed sphere.^[6]

The circumscribed sphere is the three-dimensional analogue of thecircumscribed circle.Allregular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere. The circumscribed sphere (when it exists) is an example of abounding sphere, a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it inlinear time.^[5]

Other spheres defined for some but not all polyhedra include amidsphere, a sphere tangent to all edges of a polyhedron, and aninscribed sphere, a sphere tangent to all faces of a polyhedron. In theregular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and areconcentric.^[7]

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

There are five convexregular polyhedra, known as thePlatonic solids. 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${\displaystyle M{A}_i}$ are the distances to the vertices${\displaystyle A_i}$, then^[8]

${\displaystyle 4(M{A}_1^2+M{A}_2^2+...+M{A}_n^2{)}^2=3n(M{A}_1^4+M{A}_2^4+...+M{A}_n^4).}$

Wikimedia Commons has media related toCircumscribed spheres.

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

• Elementary geometry
• Spheres

• CS1 maint: DOI inactive as of November 2024
• Articles with short description
• Short description is different from Wikidata
• Commons category link is on Wikidata