Similar Figures
Geometry & MeasurementSimilar figures have the same shape but not necessarily the same size; corresponding angles are equal and corresponding sides are proportional.
Definition
Two shapes are similar if they have the same shape but can be different sizes. All their angles match, and their sides are in the same ratio. One is like a scaled-up or scaled-down version of the other.
Example
A $3$-$4$-$5$ right triangle and a $6$-$8$-$10$ right triangle are similar. The angles are the same, and every side of the second triangle is exactly twice the first. Photographs enlarged or reduced maintain similar shapes.
Key Insight
All squares are similar to each other, and all circles are similar to each other. But not all rectangles are similar: a $2 \times 3$ rectangle and a $2 \times 6$ rectangle have the same angles but different side ratios.
Definition
Two polygons are similar if all corresponding angles are congruent and all corresponding side lengths are proportional (sharing the same ratio $k$, the scale factor). Notation: triangle $ABC \sim$ triangle $DEF$. For triangles, AA (two angles) is sufficient to prove similarity.
Example
Triangle $ABC$ with sides $5$, $7$, $9$ and triangle $DEF$ with sides $10$, $14$, $18$: the ratio is $2:1$ ($k=2$). All angles are equal. Area ratio $= k^2 = 4$, so the larger triangle has $4$ times the area of the smaller.
Key Insight
When two figures are similar with scale factor $k$, their perimeters are in ratio $k$ and their areas are in ratio $k^2$. This is why enlarging a photo by factor $2$ doubles linear dimensions but quadruples the paper area needed.
Definition
Two figures are similar if one can be mapped onto the other by a similarity transformation (a composition of a rigid motion and a dilation). Similarity is an equivalence relation coarser than congruence. In the complex plane, similarity transformations are conformal maps of the form $f(z) = az + b$ (where $a$ is nonzero), preserving angles and scaling lengths by $|a|$.
Example
All parabolas are similar: $y = x^2$ and $y = 4x^2$ are related by a scaling. More surprisingly, all conic sections of the same type (all ellipses with the same eccentricity, all parabolas) are similar, a key result in projective geometry.
Key Insight
Fractal geometry is built on self-similarity: a fractal looks similar to itself at every scale. The Sierpinski triangle is exactly similar to $3$ copies of itself scaled by $1/2$. Fractal dimension quantifies how similarity scales with complexity, generalizing the integer-valued notion of dimension.