The total angular defect is the sum of the angular defects over all polyhedron vertices of
a polyhedron, where the angular
defect 
at a given polyhedron vertex is the difference
between the sum of face angles and 
. For any convex polyhedron,
the Descartes total angular defect is
 |
(1)
|
This is equivalent to the polyhedral formula for a closed rectilinear surface, which satisfies
 |
(2)
|
A polyhedron with 
equivalent polyhedron vertices
is called a Platonic solid and can be assigned
a Schläfli symbol 
. It then satisfies
 |
(3)
|
and
 |
(4)
|
so
 |
(5)
|
