The Schläfli differential formula relates the first-order change in the volume of a polyhedron to the changes in its dihedral angles during a deformation that depends smoothly on a parameter in a space of constant curvature. For a tetrahedron in three-dimensional hyperbolic geometry, Euclidean geometry, or spherical geometry, it is
 |
where 
,
0, or 1 is the curvature, respectively, and the sum
is over the six edges 
, with edge length 
and interior dihedral angle 
.
Using exterior rather than interior dihedral angles
changes the sign convention. In the Euclidean case, the formula reduces to 
.
Akopyan and Izmestiev (2019) used the Schläfli differential formula to prove that Regge symmetry preserves the volumes of spherical tetrahedra and hyperbolic tetrahedra.
