Scale Factor
A scale factor is the ratio between corresponding measurements of a scaled figure and an original figure, indicating how much the figure has been enlarged or reduced.
Formula
\text{scale factor} = \frac{\text{new length}}{\text{original length}}
Definition
A scale factor is the number you multiply measurements by to make something bigger or smaller while keeping the same shape: a scale factor greater than $1$ makes things bigger, less than $1$ makes things smaller. Formally, the scale factor $k$ of a dilation is the ratio new length/original length; all lengths multiply by $k$, all areas by $k^2$, and all volumes by $k^3$, since similar figures have equal corresponding angles and sides in a constant ratio. A scale factor $k$ defines a dilation $D_k(v) = kv$, a linear transformation with eigenvalue $k$; in fractal geometry, self-similar sets are built from iterated function systems where each contraction has a fixed scale factor, and the Hausdorff dimension of such a fractal is determined by solving $\sum r_i^d = 1$ for its scale factors $r_i$.
Example
A drawing of a bedroom made with scale factor $1/12$ shows a $12$-foot wall as $1$ foot ($12$ inches) on paper; a scale factor of $2$ would make everything twice as big. Triangle A with sides $3$, $4$, $5$ cm scaled to similar Triangle B with sides $9$, $12$, $15$ cm has scale factor $9/3 = 3$, so if A has area $6$ cm$^2$, B has area $6 \times 3^2 = 54$ cm$^2$. The Sierpinski triangle uses three contractions each with scale factor $1/2$; its fractal dimension $d$ satisfies $3 \times (1/2)^d = 1$, giving $d = \log(3)/\log(2) \approx 1.585$, neither a curve nor a surface.
Key Insight
Scale factors keep the shape the same while changing the size: maps, model cars, and blueprints all use scale factors, and the resulting shape is "similar," same angles, proportional sides, just a different size. The area and volume scaling rules ($k^2$ and $k^3$) have surprising practical consequences: doubling the dimensions of a structure uses $4$ times the surface material but creates $8$ times the enclosed volume, the "square-cube law," which is part of why larger animals can be more efficient in some ways than smaller ones. This generalizes far beyond simple shapes: the Hausdorff dimension formula for self-similar fractals extends the scale-factor/area/volume relationships of Euclidean geometry to non-integer dimensions.