Similarity (geometry)

Lore & Background

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not generally similar to each other, rectangles are not generally similar to each other, and isosceles triangles are not generally similar to each other, because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles.

Reader's Guide

Similarity is a central concept in Euclidean geometry, with several criteria for triangles. Two triangles are similar if and only if corresponding angles have the same measure, which implies that the lengths of corresponding sides are proportional. This is known as the AAA similarity theorem. Other criteria include: any two pairs of angles are congruent; all corresponding sides are proportional; or any two pairs of sides are proportional and the included angles are congruent (SAS similarity criterion). Two congruent shapes are similar, with a scale factor of 1, though some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry, including the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem, and the Pythagorean theorem. They also provide the foundations for right triangle trigonometry. In hyperbolic geometry, where Wallis's postulate is false, similar triangles are congruent.

Did You Know?

• All circles, all squares, and all equilateral triangles are similar to each other.
• Two triangles are similar if any two pairs of angles are congruent, which in Euclidean geometry implies all three angles are congruent.
• The SAS similarity criterion requires two pairs of sides to be proportional and the included angles to be congruent.
• In hyperbolic geometry, similar triangles are congruent.

The Core Idea: Shape Preserved Under Uniform Transformation

In Euclidean geometry, similarity captures the idea that two figures share an identical shape even when their sizes differ. Formally, one figure can be transformed into the other through a uniform scaling—either enlarging or reducing—combined with any combination of translation, rotation, and reflection. This means that if you take one object, resize it proportionally, reposition it in space, spin it, and possibly flip it, it will land exactly on top of the other object. Equivalently, each figure is congruent to a uniformly scaled copy of its counterpart. The key constraint is that the scaling must be uniform: every dimension changes by the same factor, preserving the internal proportions that define the shape. This distinguishes similarity from mere proportionality in a single direction, since all linear measurements scale together. The definition accommodates mirror images as well, so a left-handed and right-handed version of the same shape still count as similar.

Universally Similar vs. Generally Dissimilar Families

Some families of shapes are universally similar to one another, while others are not. Every circle, every square, and every equilateral triangle shares the same shape regardless of size, so any two members of these families are automatically similar. In contrast, ellipses, rectangles, and isosceles triangles are not generally similar to one another. Two ellipses can differ in their width-to-height ratio, two rectangles can have different length-to-breadth proportions, and two isosceles triangles can possess different base angles. These varying internal ratios or angle measures mean that no uniform scaling can map one onto the other. The distinction highlights that similarity is not a blanket property of a shape's name or category; it depends on whether all defining proportions are fixed. A square's four equal sides and four right angles lock in its shape completely, whereas a rectangle's angles are fixed but its side ratio is free, leaving room for dissimilarity.

Triangle Similarity Criteria: AAA, SSS, and SAS

For triangles specifically, several equivalent criteria determine whether two are similar. The AAA (angle-angle-angle) theorem states that if all three corresponding angles are congruent, the triangles are similar, and their corresponding sides must be proportional. In practice, verifying just two pairs of equal angles suffices, because the third pair follows automatically in Euclidean geometry. Another criterion requires that all three pairs of corresponding sides are proportional, which is equivalent to saying one triangle is a uniform enlargement—or its mirror image—of the other. A third, the SAS (side-angle-side) similarity criterion, demands that two pairs of sides are proportional and the included angles between them are congruent. These mnemonics—AAA, SSS, SAS—help students remember which elements to check. Because any one of these conditions guarantees the others, authors often simplify the definition of similar triangles to require only angle congruence, relying on the theorem that equiangular triangles necessarily have proportional sides.

Congruence as a Special Case and the Textbook Exception

Congruence and similarity are closely related but not identical concepts. Two congruent figures are, by definition, similar with a scale factor of exactly one, since they already coincide after translation, rotation, or reflection without any resizing. However, some school-level textbooks deliberately exclude congruent triangles from their working definition of similar triangles, insisting that the two figures must differ in size to qualify. This pedagogical choice creates a subtle tension: mathematically, congruence is a special case of similarity, yet in certain curricula the two terms are treated as mutually exclusive categories. The broader geometric definition, as stated in Euclidean geometry, does not impose this restriction—any two figures related by uniform scaling, including a scale of one, are similar. This distinction matters when reading proofs or textbook exercises, where the author's convention may silently assume that similar means similar but not congruent. Understanding both the formal definition and the pedagogical variant prevents confusion when encountering problems that hinge on whether equal-sized figures count.