Cuboctahedron

A cuboctahedron can be placed in a sphere (inscribed), so that each of its vertices will touch the inner wall of the sphere.

The radius of the described sphere of the cuboctahedron

R=a

where a is the side length.

It is not possible to place the sphere inside the cuboctahedron so that it touches all sides simultaneously.
To fulfill this condition, the sphere must touch all squares and triangles.
The distances from the center of the sphere to the middle of the triangles and to the middle of the squares are different.

The surface area of the cuboctahedron.

The surface area can be defined by the following formula:

The volume of the cuboctahedron is determined by the following formula:

A cubooctahedron can be divided by a plane into two equal parts.

In this case, the cross section will have the shape of a regular hexagon.
The following four cases of section the cubooctahedron are possible:

The famous mathematical problem "On the Seven Königsberg Bridges" or the Euler Path can be reformulated and considered on the example of the edges of a cubooctahedron.

How to consistently bypass all the edges of a cubooctahedron? One edge cannot be touched twice.

The figure shows an animated example of solving this problem.

One of the names of the task is the Euler Path. In honor of the famous mathematician Leonard Euler.