Proof of Pythagoras' theorem by tiling

Other patterns with symmetry type p4

In the figure below, the blue square is the square of the hypoteneuse of the red triangle.  The blue square is clearly equal in area to the purple plus the green square.  But the purple square is the square of one of the legs of the red triangle.  The green square is the square of the other leg.  This dissection is attributed to Henry Perigal.

In fact, the construction above is specific to the particular triangle that arises in the tiling above, but another method can be used to show the Pythagorean Theorem for any triangle.

Start with any right triangle Add the square on the hypoteneuse
Add the square on one side Extend the sides to start the grid

Now the pattern has been established.  Repeat it.

The red and yellow squares are the squares on the legs.  These 2 squares can be cut up and reassembled into a square that is the same size as the square on the hypoteneuse.  This dissection was devised by Thabit ibn Qurra around 900 A.D.

Thanks to Greg Frederickson for a correction to this page.  Greg is the author of a great book with an ingenious name:  Dissections: Plane & Fancy.

main page  Links References  Tiling from Historical Sources  Tilings arranged by symmetry group  Identifying the 17 Plane Symmetry groups

Offsite link:  40 proofs of Pythagorean Theorem

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