Applications of Mathematics, Vol. 68, No. 3, pp. 259-268, 2023

Tight bounds for the dihedral angle sums of a pyramid

Sergey Korotov, Lars Fredrik Lund, Jon Eivind Vatne

Received January 15, 2022.   Published online June 21, 2022.   OPEN ACCESS

Abstract:  We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval $(3\pi,5\pi)$. Moreover, for any number in $(3\pi,5\pi)$ there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound $4\pi$ is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.
Keywords:  pyramid; dihedral angle sum; tight angle bounds
Classification MSC:  51M20, 52B10

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Affiliations:   Sergey Korotov (corresponding author), Division of Mathematics and Physics, UKK, Mälardalen University, Box 883, 721 23 Västerås, Sweden, e-mail:; Lars Fredrik Lund, Department of Computer Science, Electrical Engineering and Mathematical Sciences, Faculty of Engineering and Science, Western Norway University of Applied Sciences, Postbox 7030, 5020 Bergen, Norway, e-mail:, Jon Eivind Vatne, Department of Economics, Norwegian Business School (BI), Kong Christian Frederiks plass 5, 5006 Bergen, Norway, e-mail:

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