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.

A family of Prytz planimeters measuring a region.