What is a scale factor?
A scale factor (often written k) is the number you multiply every length of a figure by to produce a similar, larger or smaller figure. It describes a dilation: when \(k > 1\) the figure is enlarged, when \(0 < k < 1\) it is reduced, and when \(k = 1\) the figure is unchanged. Scale factors appear in geometry, maps, blueprints, model building, and image resizing.
How to use this calculator
Enter the original dimension (a length on the starting figure) and the matching new dimension (the corresponding length on the scaled figure). The calculator divides the new value by the original value to give the linear scale factor k, then squares it for the area ratio and cubes it for the volume ratio.
The formula explained
The core relationship is $$k = \dfrac{\text{New dimension}}{\text{Original dimension}}$$ Because area is a two-dimensional measure, it scales by \(k^{2}\); volume is three-dimensional, so it scales by \(k^{3}\). For example, doubling every length (\(k = 2\)) makes the area four times larger and the volume eight times larger.
Worked example
Suppose a model is built from an original length of 4 cm and the corresponding new length is 10 cm. Then $$k = 10 \div 4 = 2.5$$ The area ratio is $$2.5^{2} = 6.25$$ meaning the new figure has 6.25 times the surface area. The volume ratio is $$2.5^{3} = 15.625$$ so it holds about 15.6 times the volume.
FAQ
What does a scale factor less than 1 mean? It means the figure shrinks. For example \(k = 0.5\) halves every length.
Why does area use k² and not k? Area depends on two dimensions multiplied together, so each is scaled by k, giving \(k \times k = k^{2}\).
Can I use any unit? Yes, as long as the original and new dimensions use the same unit, the scale factor is unitless.
