The images show the principles of a linear and a polar planimeter.
2.
The polar planimeter is restricted by design to measuring areas within limits determined by its size and geometry.
3.
For the polar planimeter the " elbow " is connected to an arm with its other endpoint O at a fixed position.
4.
For a polar planimeter the total rotation of the wheel is proportional to \ scriptstyle \ int _ t r ( t ) \, \ dot \ theta ( t ) \, dt, as the rotation is proportional to the distance traveled, which at any point in time is proportional to radius and to change in angle, as in the circumference of a circle ( \ scriptstyle \ int r \, d \ theta = 2 \ pi r ).
5.
This last integrand \ scriptstyle r ( t ) \, \ dot \ theta ( t ) can be recognized as the derivative of the earlier integrand \ scriptstyle \ tfrac { 1 } { 2 } ( r ( t ) ) ^ 2 \ dot \ theta ( t ) ( with respect to " r " ), and shows that a polar planimeter computes the area integral in terms of the " derivative ", which is reflected in Green's theorem, which equates a line integral of a function on a ( 1-dimensional ) contour to the ( 2-dimensional ) integral of the derivative.
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