Planimeter - Mathematical Derivation

Mathematical Derivation

The operation of a linear planimeter can be justified by applying Green's theorem onto the components of the vector field N, given by:

where b is the y-coordinate of the elbow E.

This vector field is perpendicular to the measuring arm EM:

and has a constant size, equal to the length m of the measuring arm:

Then:

because:

The left hand side of the above equation, which is equal to the area A enclosed by the contour, is proportional to the distance measured by the measuring wheel, with proportionality factor m, the length of the measuring arm.