How to Integrate with an AX? The Surprising Power of Planimeters – Visually Explained!

2025-06-07

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Today’s mission is to reveal a surprisingly simple trick that lets you use any old ax(e) to calculate the area of plane shapes. More broadly, we’ll explore planimeters — mechanical instruments that compute area by tracing a shape’s edge. Real mathematical magic in action!

00:00 Intro (a mathematical fairy tale)

07:16 Cool machines (history and types of planimeters)

09:05 Green's theorem (the most common high-level calculus explanation)

13:05 Easy peasy (the easiest and at the same time pretty rigorous explanation)

20:30 Full sweep (the best but somewhat handwaivy explanation)

26:50 AX (making sense of the ax planimeter)

31:45 Cute stuff

33:26 Thank you!

Things to check out:

Probably the best exposition of the handwaivy part of this video.

Robert L. Foote & Ed Sandifer, Area Without Integration: Make Your Own Planimeter, https://persweb.wabash.edu/facstaff/footer/papers/foote.sandifer.reprint.pdf

In general, check out Robert L. Foote's website, lots of planimeter and other gold :)

Robert L. Foote, How planimeters work, https://tinyurl.com/yc53vzan

https://persweb.wabash.edu/facstaff/footer/HOME.HTM

Richard Courant textbook

Differential and Integral Calculus (Volume II)-Blackie & Son, 1936

Nice exposition of why area = arm x roll starting with tracing a rectangle and a linear planimeter.

Paul Kunkel, The planimeter, https://web.archive.org/web/20021201115325/http://www.nas.com/~kunkel/planimeter/planimeter.htm

One of my favourite serious maths popularisations. My video follows the chapter on planimeters quite closely.

John Bryant and Christopher Sangwin, How round is your circle, Princeton Uni Press, 2011.

Moment planimeters, a really really nice twist to the planimeter story.

Christopher Sangwin, Locating the centre of mass by mechanical means

https://tinyurl.com/mr2dje5

That's the nice survey of maths machines that I mentioned in this video.

Fr. A Willers, Mathematische Maschinen und Instrumente, Akademie Verlag, Berlin, 1951.

Various takes on the Green's theorem explanation of planimeter math(s).

Tanya Leise, As the Planimeter’s Wheel Turns: Planimeter Proofs for Calculus Class, https://tleise.people.amherst.edu/HomePage/LeisePlanimeter.pdf

Oliver Knill, The Planimeter and the Theorem of Green

https://people.math.harvard.edu/~knill/teaching/math21a2000/planimeter/index.html

Bill Casselman and John Eggers, The Mathematics of Surveying: Part II. The Planimeter

https://www.ams.org/publicoutreach/feature-column/fcarc-surveying-two

Stan Wagon, Mathematica in action, Springer, 2010

https://www.amazon.com/Mathematica%C2%AE-Action-Problem-Visualization-Computation/dp/0387753664

Stan Wagon, Polar planimeter Wolfram demonstration project

Some animations in this video were produced using a customised version of the code underlying this demonstration project.

https://demonstrations.wolfram.com/ThePolarPlanimeter/

Nitty-gritty analysis of the hatchet planimeter.

Theory of the hatchet planimeter, Simon Farthing

https://www.ias.ac.in/public/Volumes/sadh/008/04/0351-0359.pdf

Bicycle tyre tracks, a different angle from which to look at the hatchet planimeter.

Robert Foote, Mark Levi, and Serge Tabachnikov, Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture

An old newpaper article by Marty and me about hatchet planimeters.

Marty Ross and Burkard Polster, Hatcheting a plan. The AGE, https://www.qedcat.com/archive_cleaned/161.html

Calculus without calculus

https://en.wikipedia.org/wiki/Visual_calculus

The book I mentioned at the end.

Tom Apostol and Mamikon Mnatsakanian. New Horizons in Geometry. Mathematical Association of America, 2012

The Haff company's website

https://www.haff.de

The manual for using my analog polar planimeter.

https://www.haff.de/wp-content/uploads/2018/05/manual317_315_313.pdf

An old planimeter applet (you need to figure out a way to run it, I use CheerpJ Applet Runner. Use at your own risk! :)

https://web.archive.org/web/20161011062058/http://www.leinweb.com/snackbar/planimtr/planimtr.htm

T-shirt: Label your axes https://www.teepublic.com/t-shirt/12403182-label-your-axes-funny-math

Soundtrack: Hourglass by Yehezkel Raz and The Prophets Song by Ian Post

Enjoy!

Burkard