Constructing a Rectangle with the Same Area as a Pentagon

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This Demonstration shows how to construct a rectangle with the same area as that of an arbitrary pentagon . You can drag the vertices of the pentagon.


Draw the diagonal and a parallel to meet at . The triangles and have the same base and the same altitude, so their areas are equal.

Remove from and replace it with to get the quadrilateral , which has the same area as .

Similarly, replace by so that has the same area as , which has an area equal to the area of .

Finally, replace by a rectangle with the same area.


Contributed by: Izidor Hafner (May 2017)
Open content licensed under CC BY-NC-SA



This problem is taken from [1, p. 312], which is a translation from a textbook coauthored by Tarski. The explanation is taken from [1].


[1] A. McFarland, J. McFarland and J. T. Smith, eds., Alfred Tarski: Early Work in Poland—Geometry and Teaching, New York: Birkhäuser, 2014.

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