Similarity Ratio
The similarity ratio is the constant ratio of corresponding side lengths between two similar figures.
Formula
k = \dfrac{\text{corresponding side of image}}{\text{corresponding side of original}}
Definition
The similarity ratio (also called the scale factor $k$) is the number you multiply every side of one shape by to get the corresponding side of a similar shape, found by dividing any side of the image by the corresponding side of the original: $k = \dfrac{\text{corresponding side of image}}{\text{corresponding side of original}}$. For two similar figures, lengths and perimeters scale by $k$, areas scale by $k^2$, and volumes of similar 3-D solids scale by $k^3$. Formally, $k$ is the eigenvalue of the similarity transformation $T(x) = kRx + t$ (where $R$ is a rotation/reflection matrix and $t$ a translation vector); for similar polygons, $k = \sqrt{\text{Area}_{image} / \text{Area}_{original}}$, and in complex notation a direct similarity is $f(z) = az + b$ with $|a| = k$.
Example
A small triangle with sides $2$, $3$, $4$ and a large similar triangle with sides $6$, $9$, $12$ have similarity ratio $3$: every side of the big triangle is $3$ times the matching side of the small one. A model car with similarity ratio $1:18$ to the real car, if the real car is $4.5$ m long, gives a model $4.5/18 = 0.25$ m long, and if the real hood has area $2$ m$^2$, the model's hood has area $2/(18^2) = 0.0062$ m$^2$. For similar triangles with areas $25$ cm$^2$ and $100$ cm$^2$, the similarity ratio is $\sqrt{100/25} = 2$.
Key Insight
The similarity ratio affects lengths, perimeters, and areas differently: a ratio of $3$ makes lengths $3$ times bigger, perimeters $3$ times bigger, but areas $9$ times bigger. An architect's $1:100$ scale model has areas $1:10{,}000$ of the real building and volumes $1:1{,}000{,}000$, which is why small models look disproportionately tiny when you try to imagine the real structure. This length-area-volume scaling as $k^1$, $k^2$, $k^3$ is the basis of dimensional analysis in physics, where physical quantities scale predictably under changes of scale.