A mechanical device used for calculating area on an arbitrary shape.

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This type of planimeter was used to calculate area on a mathematical curve, boundary of a country or measure and increase efficiency of machines for instance. There are two type of planimeters, those which work with precisely integrator with a mechanical device, and those which estimate the area with a approximation. The orthogonal planimeter is a first type because it has a real integrator system.

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The planimeter is composed of a frame, an axle with wheels linked to the rotating yellow plan, a polar arm which moves the position on the small wheel and two dials to measure the area. The functioning of the mechanism is based on the principle of the wheel/disc integrator, as shown in the figure below.

To obtain the area on the surface, the mechanical device makes an integration of the surface below the curve. To make this integration, the curve is modeled as a standard x,y mathematical curve. On the figure on the top, x is rotation of the yellow wheel disc, also linked to the wheel axle, therefore when the planimeter goes forward, the disc rotates. Y value is the position of the small wheel on the disc, this position is modified on the planimeter by the angular position of the polar arm. And to finish, the product of this two coordinates is obtain on the axle on the disc, the integration process is recovered on the rotation on the small wheel axle equal to ∫y dx / (radius of the disc).

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As always, the measure doesn’t depend of the shape neither the orientation of the shape, it works with concave and convex shape and is more accurate than the pritz planimeter. To obtained good accuracy, I have built the planimeter with minimize the deadpoint. For that, I have used gearing with slipping wheels instead of bevel gears which have lot of inversion deadpoint. Therefore the displacement of the planimeter is accurately transmited to the wheel disc which drives after the cursor on the dials. As a consequence, the planimeter is more accurate and have bigger range,  also, it is linear on all range, which is the best enhancement . Here is a small comparison :

  • Smallest measure : 5 cm² / pritz : 25 cm²
  • Biggest measure : 400 cm² / pritz : 150 cm²
  • Accuracy : 2 cm² / pritz : 10cm²
  • Linearity : Yes / pritz : No

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The best way to understand how does it work is to watch the video :

I have of course made the building instructions, you can download it as PDF or by browsing the gallery :

Downloading the high definition PDF building instructions. Lego Orthogonal Planimeter Nico71 – 6 mo, PDF

Want to check before if you can build the Lego Orthogonal Planimeter ? The planimeter has been indexed to Rebrickable.com, the website which matchs your parts with mocs !

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