Polar Planimeter

An analog integration device used to measure areas.

Mode of Use:

On a perfectly flat surface and holding the base fixed to it with one hand, carefully trace the perimeter of the surface you want to measure. Every integer unit of the measuring wheel represents 10 cm². If your measurement exceeds 100 cm², take note of the other wheel. Make sure the paper doesnt move and that the measuring wheel rolls without sliping (in the direction the wheel is able to roll, that is). You can hope to reach a precision of around 5%, depending on several factors, including all of the above and below (Operation and assembly).

Why Does it Work:

An application of Stokes' Theorem. You can see a proof here: https://sites.icmc.usp.br/szani/sma332/material/PlanimetersCalculusRevised.pdf

Extra Materials:

• 2.5 mm rod

• Cyanoacrylate adhesive

• Petroleum jelly

Cut the rod to these lengths: 60 mm, 19 mm, 19 mm, 14 mm, 14 mm.

Tools You Might Need:

• Small file

• Hand drill

• Permanent Marker

• Rotary tool (depending on the material of your rod)

No supports are needed.

You might need to file certain surfaces and seams in order to get freely moving parts. A 2.5 mm drill bit will help you make the rod pieces fit. I used a 2.8 mm drill bit on the measuring wheel to allow it to roll freely. If the turns counter gear suffers from elephant's foot, you will need to file that down too.

A final measure to allow for soft motion would be to add small quantities of petroleum jelly to the surfaces where the measuring wheel makes friction. I avoided big clearances in a maybe foolish attempt to allow for a somewhat precise instrument.

You might want to paint the numbers and tick marks of the measuring wheel. The photographed example shows a combination of filament change and permanent marker use.

The attached PDF file shows the assembly.

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