Prytz Planimeter Animations
Here is a planimeter of length 4 measuring an ellipse with major radius 2 and minor radius 1.  The arc-length between the initial and final positions of the chisel point (the red arc) is about 1.86, and so the planimeter's approximation of the area is about 7.44.  The true area is, of course, 2 pi, or about 6.28, and so the relative error is about 18%.  Measuring the ellipse with a longer planimeter would give a more accurate result.
Main Prytz planimeter page
Pictures of Prytz Planimeters
How a Prytz planimeter measures area
More Animations    Click on each picture to see the animation.