how to find center/radius of a sphere Say you have an irregular tetrahedron, but you know the (x,y,z) coordinates of the four vertices; is there a simple formula for finding a sphere whose center exists within the tetrahedron formed by the four points and on whose surface the four points lie?

The formula of a sphere is of the form : $$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2$$
Where $(x_0,y_0,z_0)$ is the centre of the sphere and $R$ is the radius.
If you know the four points of your tetrahedron, they must all satisfy the above equation so solving the system of four (second degree) equations with four unknowns ($x_0, y_0, z_0$ and $R$) is the way to go and gives you the centre of the sphere as well as the radius.
Beware, the above equations can give you more than one solution as they are second order.
There might be a magic formula where you just plug it in but I think it is very ugly assuming it exists. Best way would be to solve the system.
