Area and volume in non-Euclidean geometry

Abstract

We give an overview of old and recent results on area and volume in hyperbolic and spherical geometries. First, we present the known results about Heron's and Ptolemy's theorems. Then we present non-Euclidean analogues of Brahmagupta's theorem for a cyclic quadrilateral. We produce also hyperbolic and spherical versions of Bretschneider's formula for the area of a quadrilateral. We give hyperbolic and spherical analogues of Casey's theorem which is a generalization of Ptolemy's equation. We give a short historical review of volume calculations for non-Euclidean polyhedra. Then we concentrate on recent results concerning Seidel's problem on the volume of an ideal tetrahedron, Sforza's formula for a compact tetrahedron in ${\mathbb{H}}^3$ or ${\mathbb{S}}^3$ and volumes of non-Euclidean octahedra with symmetries.