similarity in geometry

Two figures K and K in a Euclidean planeMathworldPlanetmath or space ( are  similarMathworldPlanetmathPlanetmathPlanetmath  iff there exists a bijectionMathworldPlanetmath f from the set of points of K onto the set of points of K such that, for any P,QK, the ratio


of the lengths of the line segmentsMathworldPlanetmath PQ and PQ is always the same number k, where P=f(P) and Q=f(Q).

The number k is called the ratio of similarity or the line ratio of the figure K with respect to the figure K (N.B. the in which the figures are mentioned!).  The similarity of K and K is often denoted by




  • Not all rectanglesMathworldPlanetmathPlanetmath are similar.

  • Not all rhombi are similar.

  • Not all rectangular prisms are similar.

  • Not all ellipses are similar.

  • Not all ellipsoidsMathworldPlanetmathPlanetmath are similar.

  • Not all trianglesMathworldPlanetmath are similar.


  • The corresponding angles (consisting of corresponding points) of two similar figures are equal.

  • The lengths of any corresponding arcs of two similar figures are proportional in the ratio k.

  • The areas of two similar regions are proportional in the ratio k2 when k is the line ratio of the regions.

  • The volumes of two similar solids are proportional in the ratio k3 when k is the line ratio of the solids.


  • In any Euclidean space E, the relationMathworldPlanetmath ( of similarity (denoted ) on the set of figures in E is an equivalence relationMathworldPlanetmath.

  • If one pair of corresponding line segments in the similar figures K and K are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal : They are congruent ( (KK).

Title similarity in geometry
Canonical name SimilarityInGeometry
Date of creation 2013-03-22 17:08:48
Last modified on 2013-03-22 17:08:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Definition
Classification msc 51F99
Classification msc 51M05
Classification msc 51-00
Synonym similarity
Synonym similitude
Related topic Homothety
Related topic ProportionEquation
Related topic HarmonicMeanInTrapezoid
Defines similar
Defines ratio of similarity
Defines similitude ratio
Defines line ratio