## Abstract

We give an overview old and recent results on areas and volumes in hyperbolic

and spherical geometries. First, we observe the known results about Heron’s and Ptolemy’s theorems. Then we present non-Euclidean analogues of the Brahmagupta’s theorem for a cyclic quadrilateral. We produce also hyperbolic and spherical versions of the Bretschneider’s formula for the area of a quadrilateral. We give hyperbolic and spherical analogues of the Casey’s theorem which is a generalization of the 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 H3 or S3 and volumes of non-Euclidean octahedra with symmetries

and spherical geometries. First, we observe the known results about Heron’s and Ptolemy’s theorems. Then we present non-Euclidean analogues of the Brahmagupta’s theorem for a cyclic quadrilateral. We produce also hyperbolic and spherical versions of the Bretschneider’s formula for the area of a quadrilateral. We give hyperbolic and spherical analogues of the Casey’s theorem which is a generalization of the 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 H3 or S3 and volumes of non-Euclidean octahedra with symmetries

Original language | English |
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Title of host publication | EIGHTEEN ESSAYS IN NON-EUCLIDEAN GEOMETRY |

Editors | Alberge, A Papadopoulos |

Publisher | EUROPEAN MATHEMATICAL SOC |

Pages | 151-189 |

Number of pages | 39 |

ISBN (Print) | 978-3-03719-196-5 |

DOIs | |

Publication status | Published - 2019 |

### Publication series

Name | IRMA Lectures in Mathematics and Theoretical Physics |
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Publisher | EUROPEAN MATHEMATICAL SOC |

Volume | 29 |

## Keywords

- CYCLIC POLYGONS
- SEIDEL PROBLEM
- FORMULA
- THEOREM