File:Invent square root of 5.svg

If the perpendicular edges of a right triangle measure 1 and 2,  then we can compute its hypotenuse length by applying the Pythagorean theorem.  The image gives us another way to compute it.  A direct similarity that${\displaystyle {\text{ multiplies areas by }}\,s\,}$ has a ${\displaystyle {\text{coefficient of }}\,{\sqrt {s}},}$  and ${\displaystyle \,s\,=\,5\,}$ in the image.

The area of each right triangle equals half an area of rectangle:

${\displaystyle 0.5\,(2\,\times \,1)\,=\,1,}$

${\displaystyle 0.5\,(2\,h\,h)\,=\,h^{\,2}.}$

The area of the enlarged triangle is 5 times the area of the initial triangle,  ${\displaystyle {\text{therefore }}\quad h^{\,2}\,=\,5\,\times \,1\,=\,5.}$