A planimeter uses Green’s theorem to measure the area of complex two-dimensional shapes. It divides the area into rectangles and uses the theorem to calculate the curved sections. A polar planimeter automates this process and measures the rotations of a wheel to determine the area covered. It is commonly used on scaled representations such as maps or building plans.
A planimeter is a tool used to measure the area of a two-dimensional shape, usually too complex for manual measurement. The device mechanizes the area calculation process using a formula known as Green’s theorem. While a floor plan can be used on a small physical area itself, it is most commonly used on a scaled representation such as a map.
The key to how a planimeter works is Green’s theorem, named after the 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 computation. However, calculating the area of curved sections is more complicated. Green’s theorem uses the available information to calculate the area of these sections. While relatively simple by computational standards, Green’s theorem would be so complicated and time-consuming as to make it largely impractical for human computation of an area.
A floor plan 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 that locks in place but can rotate. The “pulse” is a moving pointer used to plot area. The “elbow” is a hinge that can move in either direction.
This configuration means that the hinge is affected by both the direction and distance the pointer travels. The planimeter will have a wheel near the “elbow” that measures this figure. This wheel will scroll or rotate, depending on which direction you move the pointer.
As the wheel spins, a connected display will measure its rotations. The number of rotations is proportional to the area covered. It is set up so that once the pointer has traveled the entire perimeter of the area, the figure on the scale will show the size of the area. This mechanical operation eliminates the need to calculate the area with Green’s theorem.
In theory, a planimeter can be used to measure full-size two-dimensional areas. In practice there is little demand for this with areas that are small enough to be practical. In reality, a planimeter is usually used on a scale drawing of the area. This can often involve measuring an area of land using a planimeter on a map. It can also be used by a draftsman on plans for a building.
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