File:2 areas multiplied by 5 under similarities from golden tilings.svg

A regular decagon with interlaced sides has its ten vertices on the sides of a convex regular pentagon:
a pentagonal tiling by ${\displaystyle \;5\times 18=90\;{\text{ isosceles triangles,}}\,}$ divided into:

${\displaystyle 5\times 11=55\;}$ acute-angled elements with only one 36 ° angle each,

${\displaystyle 5\times 7=35\;}$ obtuse-angled elements having each each two 36 ° angles.

Every tiling element of any shape is called golden triangle because of its length ratio of two sides,  equal :

either to ${\displaystyle {\text{the golden number φ}}\ =2\,\cos 36\,^{\circ }\;;}$

or to its multiplicative ${\displaystyle {\text{inverse φ}}-1={\frac {1}{\text{φ}}}\;;}$

if not equal to 1,  of course.

The triangular tilings of two or ten elements are constructed in the same way as the pentagonal tessellation. Any similar elements of tilings are congruent,  and these tilings are contructed edge‑to‑edge:  every boundary between two adjacent elements is a full edge of one and the other element.

On the left, the two triangular tilings are similar to a smallest tiling element,  because of their two 36° angles.  So their largest length ratios of two sides equal ${\displaystyle {\text{ φ, }}}$ the golden ratio.  And ${\displaystyle {\text{ φ }}}$ is the scale ratio of the direct similarity represented by the first curved arrow at the image bottom.  The elements labelled either from 1 to 5 or from 'a' to 'e',  according to their shapes,  show the second curved arrow represents a similarity that mutiplies areas by 5.  Therefore ${\displaystyle {\sqrt {5}}\;}$ is the scale ratio of this second direct similarity on the left,  which transforms the length unit ${\displaystyle {\text{into a length of }}\;{\sqrt {5}}\,{\text{ = φ + (φ -1) = 2 φ -1.}}}$

On the right, 18 triangular elements labelled either from 1 to 11 or from 'a' to 'g',  according to their shapes,  fill one fifth (${\displaystyle {\tfrac {1}{5}}}$)  of the area of the pentagonal tiling,  equal to the area of a similar pentagon having a side length of  ${\displaystyle {\text{φ. }}}$  Since ${\displaystyle {\text{ φ + 2 }}\,}$ is the side length of the pentagonal tiling,  ${\displaystyle {\text{ then φ }}{\sqrt {5}}{\text{ = φ + 2, }}}$ relation not written on the image.