# Circumscribed sphere > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Circumscribed_sphere > Markdown URL: https://mediated.wiki/source/Circumscribed_sphere.md > Source: https://en.wikipedia.org/wiki/Circumscribed_sphere > Source revision: 1300030795 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{short description|Sphere touching all of a polyhedron's vertices}} [[File:Вписанный куб.gif|right|thumb|Circumscribed sphere of a [cube](/source/cube)]] In [geometry](/source/geometry), a '''circumscribed sphere''' of a [polyhedron](/source/polyhedron) is a [sphere](/source/sphere) that contains the polyhedron and touches each of the polyhedron's [vertices](/source/Vertex_(geometry)).{{citation|title=The Mathematics Dictionary|first=R. C.|last=James |author-link=Robert C. James |publisher=Springer|year=1992|isbn=9780412990410|page=62|url=https://books.google.com/books?id=UyIfgBIwLMQC&pg=PA62}}. The word '''circumsphere''' is sometimes used to mean the same thing, by analogy with the term ''[circumcircle](/source/circumcircle)''.{{citation|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=9781466504295|page=144|url=https://books.google.com/books?id=WLAFlr1_2S4C&pg=PA144}}. As in the case of two-dimensional circumscribed circles (circumcircles), the [radius](/source/radius) of a sphere circumscribed around a polyhedron {{mvar|P}} is called the [circumradius](/source/circumradius) of {{mvar|P}},{{citation|title=Methods of Geometry|first=James T.|last=Smith|publisher=John Wiley & Sons|year=2011|isbn=9781118031032|page=419|url=https://books.google.com/books?id=B0khWEZmOlwC&pg=PA419}}. and the center point of this sphere is called the [circumcenter](/source/circumcenter) of {{mvar|P}}.{{citation|title=Modern pure solid geometry|first=Nathan|last=Altshiller-Court|edition=2nd|publisher=Chelsea Pub. Co.|year=1964|page=57}}. ==Existence and optimality== When it exists, a circumscribed sphere need not be the [smallest sphere containing the polyhedron](/source/Smallest-circle_problem); for instance, the tetrahedron formed by a vertex of a [cube](/source/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](/source/convex_hull) of a subset of the vertices of the polyhedron.{{citation | last1 = Fischer | first1 = Kaspar | last2 = Gärtner | first2 = Bernd | last3 = Kutz | first3 = Martin | contribution = Fast smallest-enclosing-ball computation in high dimensions | doi = 10.1007/978-3-540-39658-1_57 | pages = 630–641 | publisher = Springer | series = [Lecture Notes in Computer Science](/source/Lecture_Notes_in_Computer_Science) | title = Algorithms - ESA 2003: 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings | volume = 2832 | year = 2003| isbn = 978-3-540-20064-2 | url = http://www.mpi-inf.mpg.de/~mkutz/pubs/FiGaeKu_SmallEnclBalls.pdf }}. In ''De solidorum elementis'' (circa 1630), [René Descartes](/source/Ren%C3%A9_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: some [bipyramid](/source/bipyramid)s, 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 a [simple polyhedron](/source/simple_polyhedron) has a circumscribed circle for each of its faces, it also has a circumscribed sphere.{{citation|title=Descartes on Polyhedra: A Study of the "De solidorum elementis"|title-link=Descartes on Polyhedra|first=Pasquale Joseph|last=Federico|authorlink=Pasquale Joseph Federico|series= Sources in the History of Mathematics and Physical Sciences|volume=4|publisher=Springer|year=1982|pages=52–53}} ==Related concepts== The circumscribed sphere is the three-dimensional analogue of the [circumscribed circle](/source/circumscribed_circle). All [regular polyhedra](/source/regular_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 a [bounding sphere](/source/bounding_sphere), a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it in [linear time](/source/linear_time). Other spheres defined for some but not all polyhedra include a [midsphere](/source/midsphere), a sphere tangent to all edges of a polyhedron, and an [inscribed sphere](/source/inscribed_sphere), a sphere tangent to all faces of a polyhedron. In the [regular polyhedra](/source/regular_polyhedra), the inscribed sphere, midsphere, and circumscribed sphere all exist and are [concentric](/source/Concentric_spheres).{{citation|last=Coxeter|first=H. S. M.|authorlink=Harold Scott MacDonald Coxeter|title=[Regular Polytopes](/source/Regular_Polytopes_(book))|edition=3rd|year=1973|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/16 16–17]|contribution=2.1 Regular polyhedra; 2.2 Reciprocation|contribution-url=https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16}}. When the circumscribed sphere is the set of infinite limiting points of [hyperbolic space](/source/hyperbolic_space), a polyhedron that it circumscribes is known as an [ideal polyhedron](/source/ideal_polyhedron). ==Point on the circumscribed sphere== There are five convex [regular polyhedra](/source/regular_polyhedra), known as the [Platonic solids](/source/Platonic_solids). All Platonic solids have circumscribed spheres. For an arbitrary point M on the circumscribed sphere of each Platonic solid with number of the vertices n, if MA_i are the distances to the vertices A_i, then{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 11 July 2025|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}} :4(MA_1^{2}+MA_2^{2}+...+MA_n^{2})^2=3n(MA_1^{4}+MA_2^{4}+...+MA_n^{4}). ==References== {{reflist}} ==External links== {{commonscat|Circumscribed spheres}} * {{mathworld | urlname = Circumsphere | title = Circumsphere}} Category:Elementary geometry Category:Spheres --- Adapted from the Wikipedia article [Circumscribed sphere](https://en.wikipedia.org/wiki/Circumscribed_sphere) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Circumscribed_sphere?action=history)). 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