The hyperbolic polar sine is a function of an -dimensional simplex in hyperbolic space. It is analogous to the polar sine of an -dimensional simplex in elliptic or spherical space. If the edges between vertices and have length , the value of the hyperbolic polar sine of the -dimensional hyperbolic simplex in space with Gaussian curvature is given by
The hyperbolic polar sine is used in the generalized law of sines for a hyperbolic simplex.
The limit of the hyperbolic polar sine of an -dimensional hyperbolic simplex as the curvature of the space approaches zero is , where is the content of the Euclidean simplex with the same edge lengths.