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In mathematics, a hyperbolic angle is a geometric parameter that defines a hyperbolic sector. The relationship of a hyperbolic angle to a hyperbola parallels the relationship of an "ordinary" angle to a circle.
The magnitude of the hyperbolic angle is the area of the corresponding sector of the hyperbola xy = 1. This hyperbola is rectangular with a semi-major axis of 2 {\displaystyle {\sqrt {2}}} , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius 2 {\displaystyle {\sqrt {2}}}.
Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular trigonometric functions by regarding a hyperbolic angle as defining a hyperbolic triangle.The parameter thus becomes one of the most useful in the calculus of real variables.