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The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length {{radic|2}} times the circular and hyperbolic functions.
The hyperbolic angle is an [[invariant measure]] with respect to the [[squeeze mapping]], just as the circular angle is invariant under rotation.<ref>[[Mellen W. Haskell|Haskell, Mellen W.]], "On the introduction of the notion of hyperbolic functions", [[Bulletin of the American Mathematical Society]] '''1''':6:155–9, [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf full text]</ref>
The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.
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