The circumsphere of given set of points, commonly the vertices of a solid, is a sphere that passes through all the points. A circumsphere does not always exist, but when it does, its radius R is called the circumradius and its center the circumcenter. The circumsphere is the 3-dimensional generalization of the circumcircle.


The figures above depict the circumspheres of the Platonic solids.

The circumsphere is implemented in the Wolfram Language as Circumsphere[pts], where pts is a list of points, or Circumsphere[poly], where poly is a Polygon (giving a two-dimensional circumcircle) or Polyhedron (giving a three-dimensional circumsphere) object.

By analogy with the equation of the circumcircle, the equation for the circumsphere of the tetrahedron with polygon vertices (x_i,y_i,z_i) for i=1, ..., 4 is

 |x^2+y^2+z^2 x y z 1; x_1^2+y_1^2+z_1^2 x_1 y_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4 1|=0.

Expanding the determinant,



 a=|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|,

D_x is the determinant obtained from the matrix

 D=[x_1^2+y_1^2+z_1^2 x_1 y_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4 1]

by discarding the x_i column (and taking a plus sign) and similarly for D_y (this time taking the minus sign) and D_z (again taking the plus sign)

D_x=+|x_1^2+y_1^2+z_1^2 y_1 z_1 1; x_2^2+y_2^2+z_2^2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 y_3 z_3 1; x_4^2+y_4^2+z_4^2 y_4 z_4 1|
D_y=-|x_1^2+y_1^2+z_1^2 x_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 z_4 1|
D_z=+|x_1^2+y_1^2+z_1^2 x_1 y_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 1|

and c is given by

 c=|x_1^2+y_1^2+z_1^2 x_1 y_1 z_1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4|.

Completing the square gives


which is a sphere of the form


with circumcenter


and circumradius


See also

Circumcenter, Circumcircle, Circumradius, Insphere, Midsphere

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Cite this as:

Weisstein, Eric W. "Circumsphere." From MathWorld--A Wolfram Web Resource.

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