Scale Factor (Geometry)
Geometry & MeasurementA 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. A scale factor of $2$ means the new shape is twice as big. A scale factor of $1/2$ means the new shape is half as big.
Example
A rectangle is $4$ cm $\times$ $6$ cm. If you dilate it with scale factor $3$, the new rectangle is $12$ cm $\times$ $18$ cm. If the scale factor is $0.5$, the new rectangle is $2$ cm $\times$ $3$ cm.
Key Insight
Scale factor $> 1$ means enlargement. Scale factor between $0$ and $1$ means reduction. Scale factor $= 1$ means no change. Scale factor $= -1$ means same size but reflected through the center.
Definition
The scale factor $k$ of a dilation is the ratio of any length in the image to the corresponding length in the preimage: $k = \text{image length} / \text{original length}$. For $k > 1$ the image is an enlargement; for $0 < k < 1$ it is a reduction; for $k = 1$ it is the identity. Negative $k$ includes a $180$-degree rotation about the center.
Example
A map has scale factor $1:50{,}000$, meaning $1$ cm on the map $= 50{,}000$ cm $= 500$ m in reality. A $3$ cm road on the map represents $3 \times 500 = 1500$ m $= 1.5$ km in reality.
Key Insight
Scale factors appear in scale drawings, model building, photography (zoom factor), and image resizing. In all these cases, the key property is that all dimensions change by the same factor, preserving shape while changing size.
Definition
In a dilation $D(C, k)$ centered at $C$ with scale factor $k$, every point $P$ maps to $P' = C + k(P - C)$. The scale factor $k$ is the eigenvalue of the linear part of the transformation. For $k \neq 1$, the only fixed point is the center $C$. In projective geometry, dilations are special cases of projective transformations.
Example
Composing two dilations $D(C_1, k_1)$ and $D(C_2, k_2)$: the result is a dilation with scale factor $k_1 k_2$ (if $C_1 = C_2$) or a dilation with a new center (if $k_1 k_2 \neq 1$) or a translation (if $k_1 k_2 = 1$). This composition rule is important in spiral similarity problems.
Key Insight
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. Spiral similarities appear in the logarithmic spiral, the chambered nautilus shell, and the Mandelbrot set at many locations.