The vertex figure at a vertex 
of a polygon is the line segment
joining the midpoints of the two adjacent sides meeting
at 
. For a regular 
-gon with side length 
, the length 
of the vertex figure is
 |
The vertex figure at a vertex 
of a polyhedron is the polygon
whose sides are the vertex figures of the faces surrounding 
. The faces that join at a polyhedron
vertex form a solid angle whose section by the
plane is the vertex figure, as illustrated above for one vertex of the cube.
The vertex figures of the Platonic solids yield the polyhedra (with holes centered on the centroids of the original faces) have convex hulls illustrated above and summarized in the following table.
| polyhedron | convex hull of vertex figures |
| cube | cuboctahedron |
| dodecahedron | icosidodecahedron |
| icosahedron | icosidodecahedron |
| octahedron | cuboctahedron |
| tetrahedron | octahedron |
The illustrations above show the Archimedean solids, their vertex figures, and the solids obtained by taking the convex hulls of the vertex figures.
