Similarity

Two geometric figures are similar if there is a one-to-one correspondence between their points such that all corresponding angles are equal and all corresponding lengths are scaled by the same positive constant, called the scale factor. Similarity matters because it lets us reason about large or inaccessible objects (the height of a building, the distance to a star) using small, measurable ones, and it underlies the entire subject of trigonometry. A surprising fact: while corresponding lengths scale by a factor $k$, areas scale by $k^2$ and volumes by $k^3$ — so doubling every dimension of a solid multiplies its volume by eight, not by two. Similarity is the special case of congruence when the scale factor equals 1.

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.

Formal Statement

Similar Figures

Two figures $F$ and $F^{\prime }$ are similar, written $F\sim F^{\prime }$, 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

Here $PQ$ denotes the length of segment $PQ$ in figure $F$, and $P^{\prime }{Q}^{\prime }$ the length of the corresponding segment in $F^{\prime }$. When $k=1$ the figures are congruent.

Similar 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 $A^{\prime }{B}^{\prime }{C}^{\prime }$ with $A\leftrightarrow A^{\prime }$, $B\leftrightarrow B^{\prime }$, $C\leftrightarrow C^{\prime }$:

Triangle 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.

Area and Volume Scaling

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

Why It Holds / Derivation

AA implies Similarity — Triangle Sides are Proportional

We show that if two angles match, the sides are proportional.

1. Let triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ have $\angle A=\angle A^{\prime }$ and $\angle B=\angle B^{\prime }$.
2. Since the angles of any triangle sum to $180^{\circ }$, the third angles satisfy $\angle C=180^{\circ }-\angle A-\angle B=180^{\circ }-\angle A^{\prime }-\angle B^{\prime }=\angle C^{\prime }$. So all three angles match.
3. Place $A^{\prime }{B}^{\prime }{C}^{\prime }$ so that $A^{\prime }$ coincides with $A$ and side $A^{\prime }{B}^{\prime }$ lies along $AB$. Because $\angle A=\angle A^{\prime }$, side $A^{\prime }{C}^{\prime }$ lies along $AC$.
4. Because $\angle B=\angle B^{\prime }$, the line $B^{\prime }{C}^{\prime }$ 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):

1. A parallel-line argument on the third side extends this to $\frac{A{B}^{\prime }}{AB}=\frac{A{C}^{\prime }}{AC}=\frac{B^{\prime }{C}^{\prime }}{BC}=k.$ All three ratios are equal, so the triangles are similar. $■$

Area Scaling by $k^2$ for Similar Triangles

1. The area of a triangle is $\frac{1}{2}\cdot \text{base}\cdot \text{height}$.
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

1. The same argument, with an extra factor of $k$ for depth, gives volume scaling $k^3$. $■$

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.

Building 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.

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

Worked Examples

Finding a Missing Side by Proportion

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

1. Corresponding sides share this factor, so $EF=k\cdot BC=\frac{2}{3}\times 9$.
2. Compute: $EF=6$ cm.

Shadow Method — Height of a Tree

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

1. Solve: $h_{\text{tree}}=14\times \frac{1.5}{2.0}=14\times 0.75$.
2. Compute: $h_{\text{tree}}=10.5$ m.

Common Misconceptions

Connections

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.

Real-World Applications

1. Maps and scale models — A 1:25000 Ordnance Survey map reduces every ground length by 25000; architects and engineers build similar scale models of buildings and bridges.
2. Indirect measurement — Surveyors and ancient astronomers used similar triangles to find heights and distances; Thales reputedly measured the Great Pyramid by its shadow, and Eratosthenes estimated Earth's circumference using proportional reasoning about shadows.
3. Photography and screens — Resizing an image, choosing a 16:9 aspect ratio, or projecting a slide all rely on preserving similarity to avoid distortion.
4. Camera and eye optics — A pinhole camera or lens forms an image that is similar to (a scaled copy of) the object; the magnification is a scale factor governed by similar triangles.
5. Engineering scaling laws — The square–cube law ($k^2$ surface area vs. $k^3$ volume) dictates why large structures need proportionally thicker supports and why heat dissipation favors small bodies.
6. Fractals and self-similarity — Coastlines, ferns, and snowflakes exhibit similar structure at different scales, the basis of fractal geometry used in computer graphics and antenna design.
7. Trigonometric tables and navigation — Every use of $\sin$, $\cos$, $\tan$ in surveying, GPS triangulation, and astronomy ultimately rests on the similarity of right triangles.

Historical Development

• 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.

Subtleties & Limits

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.

See Also