Pythagorean Theorem: illustration with sand

The Pythagorean theorem is prob­ably taught to school­children in all coun­tries and many science museums have an exhibit illus­trating it, often with water rather than sand.

In both cases it is impor­tant to make sure that the squares are filled completely — we are talking about equality of areas! Tech­ni­cally and visu­ally, strips between squares with “reserves” of sand or water, closed off from the observer’s view, can help.

Note another impor­tant detail. The square built on the hypotenuse is divided into two parts by the contin­u­a­tion of the alti­tude of the right triangle, dropped from the top of the right angle. It turns out that the smallest of the resulting rectan­gles is equal in area to the square built on the smaller cathetus, and the larger is equal to the square built on the larger cathetus.

Tech­ni­cally, this allows you to work with smaller volumes and, there­fore, with fewer errors. And most impor­tantly, this obser­va­tion allows constructing not illus­tra­tions, but real proofs of the Pythagorean theorem like “Look!”. And in essence it is similar to the one given in the “Ele­ments”. These beau­tiful and elemen­tary proofs may be seen here: Pythagorean Theorem: Euclid's proof page.

Other models in “Pythagorean theorem”