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Scale Factor Calculator

Enter the original and new lengths to find the scale factor.

How it is calculated

Enter the same kind of length measured on the original figure and on the new, similar figure, in one unit, for example both in centimetres. The linear scale factor is k = new ÷ original. Lengths scale by k, areas by k² and volumes by k³, and the tool shows all three to four decimals. A factor above 1 means an enlargement, a factor between 0 and 1 a reduction. The original length cannot be 0. The tool does not stop you entering a negative length: a negative new length gives a negative k and k³, while k² stays positive. The result is only meaningful for similar figures, where every length changes by the same factor.

Worked example

A model part is 8 cm long and the enlarged copy is 12 cm long. Scale factor: 12 ÷ 8 = 1.5000. Area factor: 1.5² = 2.2500, so every surface area is 2.25 times larger. Volume factor: 1.5³ = 3.3750, so the volume, and for the same material the mass, is 3.375 times larger. Enlarging the part by only 50% in length therefore more than triples its volume.

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Common mistakes

  • Applying the linear factor to areas and volumes. If lengths are 1.5 times larger, areas are 2.25 times and volumes 3.375 times larger, not 1.5 times.
  • Dividing the wrong way round. New ÷ original gives the factor from original to new; swapping them gives the inverse, 0.6667 here.
  • Mixing units, such as the original in inches and the new in centimetres. The ratio is only valid when both are in the same unit.
  • Comparing lengths that are not corresponding. The two measurements must be the same edge or distance on both figures.
  • Using it for figures that are not similar. A rectangle stretched more in width than in height has no single scale factor.

About this calculator

When two figures are similar, every linear dimension of one is the other's scale factor k times larger or smaller (k = new length / original length). Because area scales with the square of every linear dimension and volume with the cube, the corresponding area factor is k² and the volume factor is k³ — doubling a shape's length quadruples its area and multiplies its volume by eight.

Architects and model makers use this when scaling a blueprint or building a scale model, students use it to solve similar-triangle and similar-solid geometry problems, and engineers use it when resizing a 3D-printed part or a drawing while keeping proportions correct.

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