Quadrilaterals and Bretschneider's Formula

Abstract

We all learn at a young age how to calculate areas of squares or rectangles, but areas of irregular quadrilaterals are seldom mentioned. However, irregular quadrilaterals often appear in the real world. For instance, in a house, computing the area of walls, floors, or ceilings might be necessary to determine the amount of material needed for painting, tiling, and so on. Knowing the areas of backyards or fields is necessary for planting and computing yields. City personnel must determine areas of properties to compute prices and taxes. In our own research, computing areas of irregular quadrilaterals was needed for the interpretation of an ancient Aztec document that discussed surveying (see Jorge y Jorge et al. 2011). In this article, we talk about a formula deduced by Bretschneider in the mid-nineteenth century that not only allows the computation of areas of planar quadrilaterals, regular and irregular, but also provides a more thorough understanding of the geometry of such figures.