A Diagrammatic Proof of The Pythagorean Theorem

Here is a nice diagrammatic proof of the Theorem of Pythagoras involving pentagons:

Pythagorean Proof with Pentagons



  1. Start with the right-angled triangle, and a square drawn on each side.
  2. Color the two smaller squares one color and the larger square a different color. Pythagoras’ theorem states that these two areas are the same.
  3. Now color three triangles the same size as the original one with each of the two colors (as illustrated).
  4. This makes two irregular, but identical, pentagons that clearly have the same area.
  5. If the two pentagons have the same area, then the two smaller squares must have the same area as the larger one.


Source for this proof:
New Zealand Maths
Mathematics in the New Zealand Curriculum
http://www.nzmaths.co.nz/Measurement/Length/Pythagoras.htm

Continuing, if we separate the two pentagons in the figure above, we can easily see the relationship between the square of the binomial (a + b) and the Pythagorean Theorem:

Pythagorean Binomial Square
After separating the two pentagons, just draw two more triangles (labeled 4) at the corners to complete the squares.
The area of each triangle is ½(ab), and the area of the 4 triangles is 2ab.
Also (a + b) is the length of the side of each square,
and then we have

      ( a + b )2 = a2 + b2 + 2ab

and also

      ( a + b )2 = c2 + 2ab

Therefore, it follows that

      a2 + b2 = c2


                  

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             © Thomas M. Green, Contra Costa College