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 
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 
for 
, ..., 4 is
 |
(1)
|
Expanding the determinant,
 |
(2)
|
where
 |
(3)
|
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]](/images/equations/Circumsphere/NumberedEquation4.svg) |
(4)
|
by discarding the 
column (and taking a plus sign) and similarly for 
(this time taking the minus sign) and 
(again taking the plus sign)
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
and 
is given by
 |
(8)
|
Completing the square gives
 |
(9)
|
which is a sphere of the form
 |
(10)
|
with circumcenter
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
and circumradius
 |
(14)
|
