Scale Factors

Calculate the directed length factor

For positive geometric lengths,

A 6 cm side enlarged to 9 cm gives $k=9/6=3/2=1.5$. Every corresponding length must be multiplied by 1.5. A 4 cm side becomes 6 cm, and a 10 cm diagonal becomes 15 cm.

To reverse this scaling, use $1/k=2/3$. “Original to image” and “image to original” are reciprocal directions. Writing the named lengths above and below the fraction is a reliable way to avoid reversing them.

The ratio is dimensionless only after compatible units have been used. A 2 cm model length representing 5 m corresponds to $2/500=1/250$, not $2/5$.

One common factor is essential

A 6 by 4 rectangle can be uniformly enlarged to 9 by 6 because $9/6=6/4=1.5$. Enlarging it to 9 by 8 gives factors 1.5 and 2 in different directions. The result is distorted, not similar to the original.

Angles and corresponding length ratios remain unchanged under uniform scaling. A larger drawing that looks roughly the same is not enough evidence: compare corresponding measurements. Similarity in Geometry supplies the general geometric relationship.

Lengths and perimeters scale by k

Because every side is multiplied by the same $k$, their sum is also multiplied by $k$:

The 6 cm by 4 cm rectangle has perimeter $2(6+4)=20$ cm. Its 9 cm by 6 cm image has perimeter $2(9+6)=30$ cm, exactly $1.5\times20$.

This applies to any corresponding total length, including an outline composed of straight and curved portions, provided the entire figure is scaled uniformly. A changed border thickness or an added fixed margin is a separate operation, not part of the same scaling rule.

Why area scales by the square

A rectangle’s two independent length directions are both multiplied by $k$:

The original area is $6\times4=24$ cm²; the image area is $9\times6=54$ cm². The area ratio is $54/24=2.25=1.5^2$, not 1.5.

The same squared factor holds for uniformly scaled plane regions generally. Small squares covering the original become squares with both side lengths scaled. Each new covering square has $k^2$ times the old area. This explains why scaling a shape’s side lengths by 2 produces four times the area.

Volume scales by the cube

For similar solids scaled uniformly in all three dimensions,

A box with internal dimensions 6 cm by 4 cm by 1 cm has volume 24 cm³. At factor 1.5, its corresponding dimensions are 9 cm, 6 cm and 1.5 cm, giving 81 cm³. The ratio is $81/24=3.375=1.5^3$.

The internal dimensions must actually scale. Keeping a wall thickness fixed while changing the outer dimensions does not generally scale the internal capacity by $k^3$. Likewise, stretching only the height by 1.5 multiplies a prism’s volume by 1.5, not by $1.5^3$.

Recovering a length factor from area or volume

If similar figures have image-to-original area ratio $49:25$, the positive length factor is

An original 10 cm length therefore becomes 14 cm. Taking the area factor itself would overstate every new length.

If similar solids have volume ratio $27:8$, the length factor is $\sqrt[3]{27/8}=3/2$. Their surface-area ratio is consequently $9:4$. The measurements determine one another only because similarity has been established.

Equal areas alone do not establish similarity. Many differently shaped regions have equal area and unrelated side ratios.

Successive percentage resizings

Successive length factors multiply. Enlarging by 20% and then reducing the resulting length by 20% gives

The final length is 96% of the original, not 100%. The corresponding area factor is $0.96^2=0.9216$. The final area is 92.16% of the original.

To undo an enlargement by 25%, multiply by $1/1.25=0.8$, a 20% reduction. The percentage used to reverse a change refers to the new whole. This connects scaling to Percentage rather than to subtraction of two percentages.

Signed dilation factors

Some coordinate-dilation conventions permit a negative factor. For a centre $C$ and point $P$,

A negative $k$ places the image on the opposite side of the centre. Lengths scale by $|k|$, areas by $k^2$, and ordinary unsigned volumes by $|k|^3$. A factor of $-2$ therefore doubles lengths; it does not create negative lengths.

The factor $k=0$ collapses the figure to the centre and is not a similarity. This page uses positive length factors for ordinary resizing; Dilation covers the point transformation in more detail.

Related topics

Apply the factors in Scale Drawings and Maps. Dimensional Analysis explains why square and cubic unit conversions also need powers of the length conversion factor.