A planimeter is an instrument used to measure the area of a two-dimensional shape, usually one which is too complex for manual measurement. The device mechanises the process of calculating the area using a formula known as Green’s theorem. While a planimeter can be used on a small physical area itself, it’s more commonly used on a scale representation such as a map.
The key to a planimeter’s operation is Green’s theorem, named after British mathematician George Green. As a somewhat simplified explanation, using the theorem works by dividing an area into as many rectangles as will fit. The remaining areas will consist of the straight edge of a rectangle and a curved edge along the outside of the area.
Working out the areas of the rectangles would be simple enough for human calculation. However, working out the area of the curved sections is more complicated. Green’s theorem uses the available information to calculate the area of these sections. While it’s comparatively simple by calculus standards, Green’s theorem would be so complicated and time-consuming to use as to make it largely impractical for human calculation of an area.
A planimeter automates this process. In its most common form, a polar planimeter, it resembles a human arm from shoulder to wrist. The “shoulder” is usually a metal cylinder which fixed in position but can rotate. The “wrist” is a movable pointer used to trace around the area. The “elbow” is a hinge which can move in both directions.
This set-up means that the hinge is affected by both the direction and distance that the pointer travels. The planimeter will have a wheel near to the “elbow” which measures this data. This wheel will either slide or rotate, depending on the direction the pointer travels.
As the wheel rotates, an attached display will measure its rotations. The number of rotations is proportional to the area covered. It’s set up in such a way that once the pointer has travelled the entire perimeter of the area, the figure on the scale will display the size of the area. This mechanical operation removes the need to calculate the area with Green’s theorem.
In theory, a planimeter can be used to measure life-size two-dimensional areas. In practice there is little demand for this with areas which are small enough that this would be practical. In reality, a planimeter is normally used on a scale drawing of the area. This can often involve measuring an area of land by using a planimeter on a map. It can also be used by a draftsman on plans for a building.