Scale Factor (Geometry)
A scale factor in geometry is the ratio by which all dimensions of a figure are multiplied in a dilation or scale drawing.
Formula
k = \dfrac{\text{image length}}{\text{original length}}
Definition
A scale factor tells you how much bigger or smaller a new shape is compared to the original: $k = \dfrac{\text{image length}}{\text{original length}}$, so $k > 1$ means enlargement, $0 < k < 1$ means reduction, and $k = 1$ means no change (negative $k$ includes a $180$-degree rotation about the center). In a dilation $D(C, k)$ centered at $C$, every point $P$ maps to $P' = C + k(P - C)$, so $k$ is the eigenvalue of the linear part of the transformation, and for $k \neq 1$ the only fixed point is the center $C$; in projective geometry, dilations are special cases of projective transformations.
Example
A rectangle $4$ cm $\times$ $6$ cm dilated with scale factor $3$ becomes $12$ cm $\times$ $18$ cm; with scale factor $0.5$, it becomes $2$ cm $\times$ $3$ cm. A map with scale factor $1:50{,}000$ means $1$ cm on the map $= 500$ m in reality, so a $3$ cm road on the map represents $1.5$ km. Composing two dilations $D(C_1, k_1)$ and $D(C_2, k_2)$ gives a dilation with scale factor $k_1 k_2$ (same center), a dilation with a new center (if $k_1 k_2 \neq 1$), or a translation (if $k_1 k_2 = 1$), a composition rule important in spiral similarity problems.
Key Insight
Scale factor $> 1$ means enlargement, between $0$ and $1$ means reduction, $= 1$ means no change, and $= -1$ means same size but reflected through the center; scale factors appear in scale drawings, model building, photography (zoom factor), and image resizing, always changing all dimensions by the same factor. The composition of a dilation and a rotation about the same center is a spiral similarity, which maps any shape to a similar, rotated, and scaled copy, appearing in the logarithmic spiral, the chambered nautilus shell, and the Mandelbrot set at many locations.