Scale Factor
Fractions & DecimalsA 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; a scale factor less than $1$ makes things smaller.
Example
A drawing of a bedroom is made with a scale factor of $1/12$. A wall that is $12$ feet long in real life is drawn as $1$ foot ($12$ inches) on the paper. A scale factor of $2$ would make everything twice as big.
Key Insight
Scale factors keep the shape the same while changing the size. Maps, model cars, and blueprints all use scale factors. The shape is "similar" (same angles, proportional sides), just a different size.
Definition
The scale factor $k$ of a dilation is the ratio new length/original length. For a scale factor $k$: all lengths multiply by $k$, all areas multiply by $k^2$, and all volumes multiply by $k^3$. Similar figures have equal corresponding angles and sides in a constant ratio (the scale factor).
Example
Triangle A has sides $3$, $4$, $5$ cm. Similar triangle B has sides $9$, $12$, $15$ cm. Scale factor $= 9/3 = 3$. Area of A $= 6$ cm$^2$. Area of B $= 6 \times 3^2 = 54$ cm$^2$. Volume comparison (for 3D): $k^3 = 27$.
Key Insight
The area and volume scaling rules ($k^2$ and $k^3$) have surprising practical consequences. Doubling the dimensions of a structure ($k=2$) uses $4$ times the surface material ($k^2=4$) but creates $8$ times the enclosed volume ($k^3=8$). This is why larger animals can be more efficient in some ways than smaller ones - the "square-cube law."
Definition
A scale factor $k$ defines a dilation $D_k: \mathbb{R}^n \to \mathbb{R}^n$ by $D_k(v) = kv$, a linear transformation with eigenvalue $k$ (all eigenvectors). Dilations are elements of the general linear group $GL(n, \mathbb{R})$; those with $k > 0$ form the multiplicative subgroup $\mathbb{R}^+$. In differential geometry, the conformal factor between two Riemannian metrics measures pointwise scale change. In fractal geometry, self-similar sets are defined by iterated function systems where each contraction has a fixed scale factor.
Example
The Sierpinski triangle uses three contractions, each with scale factor $1/2$. The fractal dimension $d$ satisfies $3 \times (1/2)^d = 1$, giving $d = \log(3)/\log(2) \approx 1.585$ - neither $1$ (curve) nor $2$ (surface), illustrating fractional dimension.
Key Insight
The Hausdorff dimension of a self-similar fractal is determined by its scale factors: if the IFS uses $n$ contractions with scale factors $r_1, \ldots, r_n$, the dimension $d$ satisfies $\sum r_i^d = 1$. This generalizes the scale-factor/area/volume relationships of Euclidean geometry to non-integer dimensions.