planimeters

Planimeters are instruments designed to determine the area of a two-dimensional region by tracing its boundary. The device operates on a principle of planar integration: the path traced by a stylus is converted into a reading proportional to the enclosed area. The term planimeter combines “plane” and “meter” and reflects its focus on planar surfaces typical in maps and drawings.

A typical mechanical planimeter consists of a fixed frame, a tracing point, and a linkage system connected to a recording dial or counter. The operator guides the tracing point along the boundary of the figure; the movement of the linkage, which effectively integrates the path, causes the dial to advance to the area value. Calibration ensures that the reading is in square units of the drawing.

Two principal forms have traditionally been used: polar planimeters and rectangular planimeters. Polar planimeters use a fixed pivot and a tracing arm; rectangular planimeters employ a sliding carriage and a linkage to follow the boundary. In each case the mechanism is designed so that the cumulative motion corresponds to the area enclosed by the trace.

Before the digital era, planimeters were standard tools in engineering drawing and cartography for measuring land areas on maps, plan drawings, and blueprints. They remain relevant as teaching devices and in some field applications; modern digital planimeters and software in GIS and CAD can measure areas by digitizing boundaries or by tracing them with a mouse or stylus.

Limitations include the need for a closed boundary, potential measurement errors from imperfect tracing or boundary approximation, and scale calibration requirements. Curved edges are approximated by small segments, and accuracy depends on the device construction and the operator’s steadiness.