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The present article describes the results of investigaton held with the help of imaginary realized by authors interactive material model. With the help of it they detected that circle and Reuleaux triangle are the figures of equal width; the investigators also calculated their area and perimeter of these figures and developed 3 new demonstrations of Barbier's theorem.

This article presents basic methods of areas of polygons on lattices calculating. The authors considered in detail and illustrated with examples the formula of Pick, beyond the scope of the school program.