Similarity

⚡ Key Takeaways

Similarity is the geometric relationship between two figures that have the same shape but not necessarily the same size. Two figures are similar when one can be obtained from the other by a uniform scaling (enlargement or reduction), possibly combined with rotation, reflection, or translation. Key result: corresponding angles are equal and corresponding side lengths are in a constant ratio. It applies throughout geometry, trigonometry, scale drawings, maps, and physics. Formalized by the ancient Greeks (Euclid, Elements Book VI).

💡 The Core Insight

Shape is captured by angles and ratios, not by absolute size. If you preserve every angle and multiply every length by the same number, you preserve shape exactly. This single idea — "same shape, scaled size" — converts impossible measurements into easy proportions.

DefinitionSimilar Figures

Two figures $F$ and $F'$ are similar, written $F \sim F'$ , if there exists a one-to-one correspondence between their points and a positive constant $k$ (the scale factor) such that for every pair of corresponding points the distances satisfy

1

\quad \frac{P'Q'}{PQ} = k \quad \text{for all corresponding } P,Q,

and corresponding angles are equal. Derivation hint: This is the definition; $k$ is the same constant for every pair of corresponding segments, which is what "uniform scaling" means. Here $PQ$ denotes the length of segment $PQ$ in figure $F$ , and $P'Q'$ the length of the corresponding segment in $F'$ . When $k=1$ the figures are congruent.

DefinitionSimilar Polygons

Two polygons are similar if their vertices can be paired so that: 1. corresponding angles are equal, and 2. corresponding sides are proportional. For two triangles $ABC$ and $A'B'C'$ with $A\leftrightarrow A'$ , $B\leftrightarrow B'$ , $C\leftrightarrow C'$ :

2

\quad \angle A=\angle A',\ \angle B=\angle B',\ \angle C=\angle C' \quad\text{and}\quad \frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'A'}{CA}=k.

Derivation hint: Both conditions follow directly from equation [1] applied to each side and each angle of the polygon.

TheoremTriangle Similarity Criteria

For two triangles, any one of the following conditions guarantees similarity:

AA (Angle–Angle): Two pairs of corresponding angles are equal.
SAS (Side–Angle–Side): Two pairs of corresponding sides are proportional and the included angles are equal.
SSS (Side–Side–Side): All three pairs of corresponding sides are proportional.

TheoremArea and Volume Scaling

If two similar figures have scale factor $k$ (ratio of corresponding lengths), then:

3

\quad \frac{\text{Area}'}{\text{Area}} = k^2, \qquad \frac{\text{Volume}'}{\text{Volume}} = k^3.

Derivation hint: Area is a product of two lengths, each scaled by $k$ , giving $k^2$ ; volume is a product of three lengths, giving $k^3$ .

ProofAA implies Similarity — Triangle Sides are Proportional

We show that if two angles match, the sides are proportional. 1. Let triangles $ABC$ and $A'B'C'$ have $\angle A=\angle A'$ and $\angle B=\angle B'$ . 2. Since the angles of any triangle sum to $180^\circ$ , the third angles satisfy $\angle C = 180^\circ - \angle A - \angle B = 180^\circ - \angle A' - \angle B' = \angle C'$ . So all three angles match. 3. Place $A'B'C'$ so that $A'$ coincides with $A$ and side $A'B'$ lies along $AB$ . Because $\angle A=\angle A'$ , side $A'C'$ lies along $AC$ . 4. Because $\angle B=\angle B'$ , the line $B'C'$ is parallel to $BC$ (equal corresponding angles imply parallel lines). 5. By the Basic Proportionality Theorem (a line parallel to one side of a triangle cuts the other two sides proportionally):

4

\quad \frac{AB'}{AB}=\frac{AC'}{AC}.

Derivation hint: Parallel lines $B'C'\parallel BC$ create equal ratios along the two transversals $AB$ and $AC$ . 6. A parallel-line argument on the third side extends this to $\dfrac{AB'}{AB}=\dfrac{AC'}{AC}=\dfrac{B'C'}{BC}=k.$ All three ratios are equal, so the triangles are similar. $\blacksquare$

∎

ProofArea Scaling by

k^2

for Similar Triangles

1. The area of a triangle is $\tfrac12 \cdot \text{base} \cdot \text{height}$ . 2. Under scale factor $k$ , the base becomes $k\cdot\text{base}$ and the corresponding height becomes $k\cdot\text{height}$ (heights are lengths and scale identically). 3. Therefore

5

\quad \text{Area}' = \tfrac12(k\cdot\text{base})(k\cdot\text{height}) = k^2\left(\tfrac12\cdot\text{base}\cdot\text{height}\right)=k^2\,\text{Area}.

Derivation hint: Substitute the scaled base and height into the area formula and factor out $k^2$ . 4. The same argument, with an extra factor of $k$ for depth, gives volume scaling $k^3$ . $\blacksquare$

∎

mathematician · Euclid of Alexandria | c. 300 BCE

Euclid systematized the theory of similar figures in Book VI of the Elements, building on the rigorous theory of proportion developed by Eudoxus of Cnidus (c. 408–355 BCE). Eudoxus's definition of equal ratios — capable of handling incommensurable (irrational) lengths — let Greek mathematicians treat similarity precisely without a modern number system. Book VI's propositions on proportional triangles and the construction of mean proportionals remained the standard treatment of similarity for over two thousand years and underpin all of classical trigonometry.

🧠 Intuition

Think of a photograph being enlarged on a photocopier set to 200%. Every length doubles, but nothing is distorted: a face stays a face, angles between features are untouched, circles stay circles. That is similarity in action — $k=2$ . Now notice the trap that surprises everyone: if you enlarge a photo to twice the width and twice the height, it uses four times the paper, because area depends on two lengths. Scale a model to twice the size in all three dimensions and it weighs eight times as much. This $k$ , $k^2$ , $k^3$ pattern is why large animals need disproportionately thick legs and why a kitten scaled to lion size could not support its own weight.

Figure

Two similar triangles: same angles, sides scaled by k = 2

ExampleFinding a Missing Side by Proportion

Given: Triangle $ABC \sim$ triangle $DEF$ with $AB=6$ cm, $BC=9$ cm, and $DE=4$ cm. The side $DE$ corresponds to $AB$ , and $EF$ corresponds to $BC$ . Find $EF$ .

Steps: 1. The scale factor from $ABC$ to $DEF$ is

6

\quad k = \frac{DE}{AB} = \frac{4}{6} = \frac{2}{3}.

Derivation hint: scale factor is the ratio of a pair of corresponding sides. 2. Corresponding sides share this factor, so $EF = k\cdot BC = \tfrac{2}{3}\times 9$ . 3. Compute: $EF = 6$ cm.

\boxed{EF = 6\ \text{cm}}

ExampleShadow Method — Height of a Tree

Given: A vertical stick 1.5 m tall casts a shadow 2.0 m long. At the same moment a tree casts a shadow 14 m long. Find the tree's height. The sun's rays are parallel, so the stick–shadow triangle and tree–shadow triangle are similar by AA (both have a right angle and equal sun-angle).

Steps: 1. Set up proportional corresponding sides (height to shadow):

7

\quad \frac{h_{\text{tree}}}{14} = \frac{1.5}{2.0}.

Derivation hint: similar triangles ⇒ ratio of height to shadow is the same for both. 2. Solve: $h_{\text{tree}} = 14 \times \dfrac{1.5}{2.0} = 14 \times 0.75$ . 3. Compute: $h_{\text{tree}} = 10.5$ m.

\boxed{h_{\text{tree}} = 10.5\ \text{m}}

⚠️ Common mistake

"Equal angles always mean similar — so all rectangles are similar." False. All rectangles have four right angles, yet a $1\times 2$ rectangle is not similar to a $1\times 5$ rectangle. For polygons with more than three sides, equal angles are not enough; the sides must also be in proportion. The AA criterion is a special privilege of triangles, because a triangle's shape is fully fixed once its angles are known.

⚠️ Common mistake

"If lengths double, area and volume double too." They do not. Area scales by $k^2$ and volume by $k^3$ . Forgetting this is the single most common error in similarity problems — a map at 1:50000 has features 50000 times shorter, but each map region represents $50000^2 = 2.5\times10^9$ times the ground area.

Counterexample

A rhombus and a square can have all four corresponding sides in the same ratio (even ratio 1, if congruent in side length) yet not be similar, because their angles differ. Consider a unit square (all angles $90^\circ$ ) and a unit rhombus with angles $60^\circ$ and $120^\circ$ . Every side length matches, the side ratios are all 1, but the figures are clearly different shapes. This shows that for non-triangles, proportional sides alone are insufficient — angles matter too.

connection

Similarity is the engine behind much of geometry. It is the scaled-up cousin of Congruence (similarity with $k=1$ ), and it provides the definition of the trigonometric ratios: in Trigonometry, $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ are well-defined precisely because all right triangles with a given angle $\theta$ are similar, so their side ratios depend only on $\theta$ . The Pythagorean Theorem has an elegant proof using the similar triangles formed by an altitude to the hypotenuse. In coordinate geometry, similarity transformations belong to the broader study of Geometric Transformations (dilations composed with isometries). The area- and volume-scaling laws connect to Dimensional Analysis in physics.

🔬 Where this appears

⏳ History

- c. 1650 BCE — The Egyptian Rhind Mathematical Papyrus uses ratio and proportion in slope ("seked") calculations for pyramids.

- c. 600 BCE — Thales of Miletus is credited with using similar triangles to measure the height of pyramids and the distance of ships at sea.

- c. 408–355 BCE — Eudoxus of Cnidus develops a rigorous theory of proportion valid for incommensurable magnitudes, making similarity logically sound.

- c. 300 BCE — Euclid codifies similarity in Book VI of the Elements, including triangle similarity and the construction of proportional segments.

- c. 240 BCE — Eratosthenes uses proportional shadow lengths to estimate Earth's circumference with remarkable accuracy.

- c. 150 CE — Ptolemy's Almagest applies chord (proto-trigonometric) ratios derived from similarity to astronomy.

- 1637 — René Descartes introduces coordinate geometry, allowing similarity transformations to be expressed algebraically.

- 19th century — Felix Klein's Erlangen Program (1872) reframes geometry by its transformation groups, placing similarity transformations in a precise hierarchy between congruence and affine geometry.

- 1975 — Benoît Mandelbrot coins "fractal," generalizing self-similarity to objects with non-integer dimension.

open-problem

The clean $k$ , $k^2$ , $k^3$ scaling laws assume smooth, ordinary shapes. For fractals, the notion of "length" can be ill-defined: the more finely you measure a coastline, the longer it gets, and it does not scale with a single integer power. Such objects are described by a fractional (Hausdorff) dimension $d$ , for which "content" scales as $k^d$ with $d$ non-integer. Determining the exact dimension of natural fractals, and rigorously defining measures for self-similar sets, remains an active area of research in geometric measure theory. Even classically, the deep question of which sets in the plane are similar to subsets of themselves at every scale connects to unsolved problems about self-similar tilings and aperiodic structures.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also