Similarity Ratio
Geometry & MeasurementThe 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) is the number you multiply every side of one shape by to get the corresponding side of the similar shape. If the ratio is $3$, every side is $3$ times as long.
Example
A small triangle with sides $2$, $3$, $4$ and a large similar triangle with sides $6$, $9$, $12$: the similarity ratio is $3$. Each side of the big triangle is $3$ times the matching side of the small one.
Key Insight
The similarity ratio affects lengths, perimeters, and areas differently. If the ratio is $k$, lengths scale by $k$, perimeters scale by $k$, but areas scale by $k^2$. So a ratio of $3$ makes areas $9$ times bigger.
Definition
For two similar figures with scale factor $k$ (the similarity ratio), all corresponding lengths are in ratio $k:1$. Perimeters are in ratio $k:1$. Areas are in ratio $k^2:1$. Volumes of similar 3-D solids are in ratio $k^3:1$. The ratio is found by dividing any side of the image by the corresponding side of the original.
Example
A model car has similarity ratio $1:18$ to the real car. If the real car is $4.5$ m long, the model is $4.5/18 = 0.25$ m long. The real car's hood has area $2$ m$^2$; the model's hood area is $2/(18^2) = 2/324 = 0.0062$ m$^2$.
Key Insight
Scale models use similarity ratios. An architect's $1:100$ scale model has areas $1:10{,}000$ of the real building and volumes $1:1{,}000{,}000$. This is why small models look disproportionately tiny when you try to imagine the real structure.
Definition
The similarity ratio $k$ is the eigenvalue of the similarity transformation $T(x) = kRx + t$ (where $R$ is a rotation/reflection matrix and $t$ is a translation vector). For similar polygons, $k = \sqrt{\text{Area}_{image} / \text{Area}_{original}}$. In complex notation, a direct similarity (orientation-preserving) is $f(z) = az + b$ with $|a| = k$.
Example
For similar triangles with areas $25$ cm$^2$ and $100$ cm$^2$, the similarity ratio is $\sqrt{100/25} = 2$. Every linear dimension of the larger triangle is $2$ times the smaller, and volumes of similar 3-D extensions would be in ratio $2^3 = 8$.
Key Insight
The similarity ratio connects linear, areal, and volumetric scaling via powers: length scales as $k^1$, area as $k^2$, volume as $k^3$. This power law is the basis of dimensional analysis in physics, where physical quantities scale predictably under changes of scale.