Similarity (geometry)

Overview. In geometry, two figures are called similar when one can be obtained from the other by a sequence of uniform resizing (dilation) possibly combined with rigid motions. Similar shapes share the same overall shape while allowing different sizes: corresponding angles are equal and corresponding side lengths are in constant proportion. The notation F  ∼  F′ is commonly used to indicate that figure F is similar to figure F′. Simple examples include any two circles or any two squares, and any two line segments, which are always similar because they differ only by scale.

Defining characteristics and criteria. For polygons the defining requirements are equality of corresponding angles and proportionality of corresponding sides. Triangles play a special role because fewer conditions suffice to establish similarity. The standard triangle criteria are:

• Angle-Angle (AA): two pairs of corresponding angles equal implies similarity.
• Side-Angle-Side (SAS) similarity: two side ratios equal and the included angle equal implies similarity.
• Side-Side-Side (SSS) similarity: all three pairs of corresponding sides are in the same ratio.

These triangle tests are powerful: for triangles, equal angles alone (AA) guarantee proportional sides and hence full similarity. For larger polygons both angle equality and side proportionality are generally required.

Transformations that realize similarity. A similarity transformation is any map composed of a dilation (scaling about a point), possibly followed by a translation, rotation, or reflection. In the plane these are sometimes called similitudes or similarity motions. The common scale factor (similarity ratio) measures how much lengths are multiplied when passing from one figure to the other. Orientation may be preserved by a rotation plus dilation or reversed by including a reflection.

History and context. The study of similarity goes back to ancient Greek mathematics: early theorems about proportional segments and similar triangles appear in the work attributed to Thales and are systematically treated in Euclid's Elements (Book VI). Over time similarity became central to geometric reasoning, construction, and measurement before modern analytic methods were widespread.

Uses, examples, and consequences. Similarity underlies many practical techniques: scale drawings and maps, model-making, indirect measurement (using similar triangles to find heights or distances), and photographic enlargement. Many geometric quantities behave predictably under similarity: areas scale by the square of the similarity ratio and volumes (in three dimensions) scale by its cube. Corresponding medians, altitudes and other linear measures are proportional, and corresponding angles remain equal.

Distinctions and related notions. Similarity is closely related to but distinct from congruence. Congruent figures are similar with scale factor 1 and can be matched by rotating, translating or reflecting alone; similarity allows an additional resizing. The term "similarity" also appears in other mathematical contexts with different meanings (for example, matrix similarity in linear algebra), so context matters. Classic facts include that all regular n-gons with the same number of sides are similar, and that similarity classes form a group under composition of similarity transformations.

Further reading and formal proofs of triangle criteria or of area/volume scaling can be found in standard geometry texts and educational resources. Definitions and constructions related to similar figures are elementary but have wide-ranging applications across mathematics, engineering, architecture and the physical sciences. For introductions and examples see materials linked from general polygon and triangle references, and discussions of proportional segments and similarity maps in classical treatments (proportionality, angles). Additional background on transformations and motion is available under topics for angle and rotation.