Computing Areas of Regions with Discretely Defined Boundaries

Abstract

It is well known that the area of a region in the plane can be computed by an appropriate integration arund the boundary of the region. If the boundary is defined by a sequence of points connected by straight lines (a polygon), the paarametric representation of the boundary is particularly simple, and an explicit formula for the area can be derived. Using Stokes' Theorem, this idea can be extended to derive area formulas for regions on non-planar surfaces whose bouondaries are defined by a sequence of points connected by appropriate curves. In this note, we present exact area formulas for regions in the plane and regions on the sphere whose boundaries are defined by such discrete sets of points.