A New Hyperbolic Area Formula of a Hyperbolic Triangle and Its Applications

Abstract

We study some characterizations of hyperbolic geometry in the Poincaré disk. We first obtain the hyperbolic area and length formula of Euclidean disk and a circle represented by its Euclidean center and radius. Replacing interior angles with vertices coordinates, we also obtain a new hyperbolic area formula of a hyperbolic triangle. As its application, we give the hyperbolic area of a Lambert quadrilateral and some geometric characterizations of Lambert quadrilaterals and Saccheri quadrilaterals. hyp ( ) = ∫ Ω ( ) | | .(4)