Planimeter

A planimeter is a technical drawing instrument used to measure the surface area of an arbitrary two-dimensional shape by tracing the perimeter. The precise way in which they are constructed varies, the main types of mechanical planimeter being polar; linear; and Prytz or "hatchet" planimeters. Invented by Swiss mathematician Jakob Amsler-Laffon in 1854, they can be justified by appealing to Green's theorem. Specifically,

$\oint _{S}xdy-ydx=\int _{S}\left({\frac {\partial x}{\partial x}}-{\frac {\partial y}{\partial y}}\right)\,dxdy=\int _{S}2dA$

where the left-hand side is readily mechanically calculated (note its similarity to the z-component of a vector cross product) as the contour is traced, while the right-hand side is twice the area enclosed by that contour.

The first planimeter was invented by Hermann in 1814 and there were many subsequent developments before Amsler's famous planimeter. Electronic versions also exist.

There was a book written in 1903 on the planimeter, The Polar Planimeter by J. Y. Wheatley.

External links

• Whistleralley site
• Larry's Planimeter Platter
• Wuerzburg Planimeter Page
• Robert Foote's planimeter page
• Computer model of a planimeter