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

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. 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in order to prove the dependence of Euclid’s fifth postulate on the first four ones. It was extensively studied by Nikolai Lobachevskii and Johann Bolyai. Because of this, hyperbolic geometry is also known as Bolyai-Lobachevskian geometry. The basis necessary for an analytic study of hyperbolic geometry was laid by Leonhard Euler, Gaspard Monge, and Friedrich Gauss in their investigation of curved surfaces. Later, two of the most famous analytic models of the hyperbolic geometry were built, which are known as the Klein model and the Poincaré model in the name of their inventors [1]. One can refer to [2–4] for more history about hyperbolic geometry. For a complex number , we assume that , where . For convenience, we denote by the coordinate of the point . We denote the complex plane, the unit disk, and the upper half plane by , , and , respectively. We denote by the Euclidean circle with center and radius . Let be the hyperbolic geodesic segment joining and when or , or the hyperbolic geodesic ray with one of the two points and is on or . Let be a simply connected domain of hyperbolic type and let be the hyperbolic metric of with the Gaussian curvature . In particular, the hyperbolic metric density in and with the Gaussian curvature is given by respectively. Define the hyperbolic area of a measurable subset of as Particularly, if is a measurable subset in , then The integrand at (3) is replaced with when . We also denote the Euclidean area of a measurable subset by . For any rectifiable curve in , the hyperbolic length of is If is or , then the integrands in (4) are and , respectively. The hyperbolic distance between two points is defined by where the infimum is taken over all rectifiable curves in joining and . We say that a curve is a hyperbolic geodesic joining and if, for all , it follows If there exists a polygon enclosed by hyperbolic geodesics, then we call it a hyperbolic polygon. Particularly, if takes or , then we call it a hyperbolic triangle and hyperbolic