Computing areas using Green's Theorem and a Software Planimeter

According to the Merriam-Webster dictionary, a planimeter is `an instrument for measuring the area of a plane figure by tracing its boundary line'. Even without knowing how a planimeter works, it is clear from the definition that the idea behind it is that one can compute the area of a figure just by `walking' on the boundary. For someone who has taken calculus, this immediately suggests Green's Theorem. The aim of this note is to clarify for others why this principle works. We do this by using points of view from linear algebra to elementary plane geometry in order to obtain an intuitive justification for Green's Theorem. As an application, we show how the reader can easily construct a `software planimeter'. The idea behind this is certainly not original; the formula for the area of a polygon used in this process (see Theorem 1.2 and Remark 1.5 (3)) is surely folklore for experienced programmers (1). What we would like to emphasize is, on the one hand, the beautiful interplay between various branches of mathematics and elementary computer graphics in this case. On the other hand, since our application is (in our view) particularly fun to work with, we hope that this approach can be successfully used in class as an aid in teaching Green's Theorem, or in a calculus lab, or even to show younger and more inexperienced students that sometimes deep mathematical results can be surprisingly accessible and entertaining.