The Polar Planimeter
The Polar Planimeter
The polar planimeter is a device that measures the area of a plane region by tracing out its boundary. The mechanism has two arms, a fixed anchor point, a freely moving elbow, a needle that traces the boundary of the region counterclockwise, and a wheel whose orientation is perpendicular to the elbow-to-needle arm. If the distance from the elbow to the needle is , then the net distance rolled by the wheel is the area divided by . This Demonstration shows two possible locations of the wheel: at the needle, or at an arbitrary point on the elbow-to-needle arm (which is how a real planimeter works). The inner wheel traces out a curve that is a morph between the path of the needle (the region being measured) and the path of the elbow (an arc of a circle).
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