The midpoint of the longest edge is the center of the circumscribed sphere.

There he discovered two simple occurrences in platonic solids and their circumscribed spheres.

The circumscribed sphere is the three-dimensional analogue of the circumscribed circle.

By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions.

There is therefore, as it were a circumscribed sphere in which each exercises its functions 'jure proprio'".

For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length.

The circumscribed sphere has radius (the circumradius)

In contrast, there exist polyhedra that do not have an equivalent form with an inscribed sphere or circumscribed sphere.

Civilization held nothing like this in its narrow and circumscribed sphere, hemmed in by restrictions and conventionalities.

Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used.

In fact, the more faces a Waterman polyhedron has, the more it resembles its circumscribed sphere in volume and total area.

The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.

In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric.

If the object does not have an obvious center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere.

We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide.

With each point of 3D space we can associate a family of Waterman polyhedra with different values of radii of the circumscribed spheres.

In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices.

All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere.

If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere (one that touches the truncated icosahedron at all vertices) is:

This common point is called the orthocenter, and it has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid.

The midsphere is so-called because it is between the inscribed sphere (which is tangent to every face of a polyhedron) and the circumscribed sphere (which touches every vertex).

The spherical tiling corresponding to a regular polyhedron is obtained by forming the barycentric subdivision of the polyhedron and projecting the resulting points and lines onto the circumscribed sphere.

A Kepler-Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others.

In Mysterium Cosmographicum, published in 1596, Kepler laid out a model of the solar system in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.

If you connect the centers A and B of any two cube surfaces and extend them again, such that the extended line intersects the circumscribed sphere in C, then B will divide AC according to the golden section.