2004/03/29 by Dragutin Svrtan, Svrtan, Dragutin, Darko Veljan +4 · 1 citation
Mathematics · #51M04 #51M25 #52A10 #52A38 #52B55 #Advanced Mathematical Theories #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #math.MG #msc:51M04 #msc:51M25 #msc:52A10 #msc:52A38 #msc:52B55
paper · pdf · doi:10.48550/arxiv.math/0403503
22 pages, 8 figures
arxiv created 2004/03/29 · openalex publication_date 2004/03/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give several proofs and extensions of this beautiful formula to hexagon etc. and consider special cases of affine--regular polygons. The Gauss pentagon formula is, in fact, equivalent to the Monge formula which is equivalent to the Ptolemy formula. On the other hand, we give a new proof of the Robbins formula for the area of a cyclic pentagon in terms of the side lengths, and this is a consequence of the Ptolemy formula. The main tool is simple: just eliminate from algebraic equations, via resultants. By combining Gauss and Robbins formulas we get an explicit rational expression for the area of any cyclic pentagon. So, after centuries of geometry of triangles and quadrilaterals, we arrive to the nontrivial geometry of pentagons.