Proposition: 6.20: Similar Polygons are Composed of Similar Triangles

(Proposition 20 from Book 6 of Euclid's “Elements”)

Similar polygons can be divided into equal numbers of similar triangles corresponding (in proportion) to the wholes, and one polygon has to the (other) polygon a squared ratio with respect to (that) a corresponding side (has) to a corresponding side.


Modern Formulation

The ratio of the areas of two similar rectilinear figures is proportional to the squared ratio of the lengths of corresponding sides.1

Proofs: 1 Corollaries: 1

Corollaries: 1
Proofs: 2 3 4

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016


  1. Euclid formulates this proposition generally for all similar rectilinear figures but proves it for pentagonal figures only (editor's note).