Polar and Linear Planimeters

How They Are Used
How They Work

The main working parts of polar and linear planimeters are the same.   Each has a rod, called the tracer arm, one end of which is the tracing point T, and a wheel attached to the rod with its axis parallel to the rod.  A scale attached to the wheel records how much the wheel turns.
Polar Planimeter 
Linear Planimeter
The two planimeters differ in the movement of the end of the tracer arm opposite the tracer point.  In the linear planimeter, that end is restricted to move along a straight line.  The drawing suggests that it runs along a track, but a more common way to cause this motion is to have this end attached to a set of wheels that are fixed to an axis so that the wheels turn together.  In the polar planimeter, the end opposite the tracer point is restricted to move along a circle.  This is done by making that point the hinge between the tracer arm and a secondary arm, one end of which (the pole) is fixed.

The original polar planimeter was invented in 1854 by Jacob Amsler, a Swiss mathematician and inventor of many measuring instruments.  It was so much simpler, easier to use, and more accurate than previously invented planimeters that the latter quickly became obsolete.  Further modifications of his basic design were made only to improve its accuracy.  The linear planimeter works on the same basic principle as the polar planimeter, and is simply a variation that allows the areas of long, skinny regions to be measured.

Here are some pictures of a Keuffel & Esser polar planimeter I have (model 620015). This was made by the German company Haff in about 1970.
The picture on the left shows the planimeter ready to measure the area of a region.  On the right is a disassembled view.  The tracer point is equipped with a magnifying glass to make it easier follow the curve.  The box at the left end of the tracer arm is a carriage containing the wheel and the counter with its scale.  The wheel rests lightly on the paper and can roll when the tracer point is moved.  The pole is at the top and is held in place by a weighted block that allows the pole arm to pivot.
The axis of the wheel is easily seen on the underside of the carriage.  The wheel itself (indicated by the pointer) is a steel disk attached directly to the cylinder with the scale.  The plastic wheel on the same axis that is visible in both pictures is slightly smaller than the steel one and does not contact the paper.  It is used to set the scale to zero.

This planimeter is on permanent loan from a member our biology department.  If you are looking for a planimeter, good places to check are physics, chemistry, and engineering departments.  They are almost always available at reasonable prices on eBay.

How Polar and Linear Planimeters are Used
These planimeters are used by moving the tracer point around the boundary of the region being measured.  As the tracer point moves the wheel partially rolls and partially slides on the paper, recording its motion perpendicular to the tracer arm.  It turns out that when the tracer point returns to the point where it started, the net roll of the wheel is proportional to the area of the region.  If a scale attached to the wheel is calibrated appropriately, the area can be read from the scale.

How Planimeters Work

Main planimeter page
Last update 2 January 2004
Robert Foote