As basis, the Pythagorean Theorem.
The length of quadrilateral side opposite that coincident with the triangle equals:
(√(a2+b2)) – (b/(2√(1+(b2/a2))))
(Ref. Fig. 1) A polygon, being a quadrilateral with right angle, containing an edge coincident with the side of a right triangle and having an area in accord with the square of the same is developed. Repeated for the adjacent side, each polygon is rotated about their respective vertex until coincident with the hypotenuse, their supplement forming the perimeter of an irregular pentagon with area compliant to the squared hypotenuse. A subsequent Pythagorean tiling or tessellation is also depicted. I recommend for your consideration the work of Volker Pöhls, Extended Theorem of Pythagoras.
Fig 1.
A GeoGebra animation of construction development can be found here: Dissimilar Shapes of Pythagorean Conformity, Part 1 – GeoGebra
A GeoGebra animation of tiling or tessellation can be found here: Dissimilar Shapes of Pythagorean Conformity, Part 2 – GeoGebra
