Cymra Haskell, USC

Title: The Planimeter:  A Hands-On Activity for Exploring Greens’ Theorem

Abstract: A planimeter is a mechanical device used by cartographers to measure the areas of regions on a map.   In this 2-part session we will explore how a planimeter works from an elementary standpoint and from the standpoint of Greens’ Theorem.  This is an activity I have done with K-12 teachers and could be suitable for Calculus III classes.

In this first of two parts, we will build a planimeter with Tinker Toyz and use it to measure the areas of different regions.  The physicality of building and using the planimeter will lead us to focus on the very features that are central to how the tool functions.  We will then take the planimeter apart to isolate the surveyor’s wheel and the rod.  We will decompose the motion of the rod into two different kinds of motion that cause the wheel to rotate.

In the second part we will put the planimeters back together to understand why the rotation of the wheel is proportional to the area of the region traced out. We will then see how we can explain the working of the planimeter using Greens’ Theorem.