What Is a Scale Factor?
A scale factor is the number you multiply the dimensions of one figure by to produce a similar figure. When two shapes are similar, every pair of corresponding lengths shares the same ratio — that ratio is the scale factor, usually written as k. A scale factor greater than 1 produces an enlargement; a value between 0 and 1 produces a reduction.
How to Use This Calculator
Choose what you want to find. To get the scale factor, enter the original dimension and the new (scaled) dimension — the tool computes \(k = \text{new} \div \text{original}\). To find a missing length, enter the dimension you know plus the scale factor, and it solves \(\text{new} = k \times \text{original}\) or \(\text{original} = \text{new} \div k\). The result also shows the area scale factor, which is \(k^2\).
The Formula Explained
For similar figures, corresponding sides are proportional:
$$k = \dfrac{\text{New dimension}}{\text{Original dimension}}$$Rearranging gives
$$\text{New dimension} = \text{Scale factor }(k) \times \text{Original dimension}$$and
$$\text{Original dimension} = \dfrac{\text{New dimension}}{\text{Scale factor }(k)}$$Because area depends on two dimensions, areas scale by \(k^2\), and volumes scale by \(k^3\). So doubling every length (\(k = 2\)) quadruples the area and multiplies volume by eight.
Worked Example
A photo is 4 inches wide and is enlarged so the new width is 10 inches. The scale factor is
$$k = 10 \div 4 = 2.5$$If the original height is 6 inches, the new height is
$$2.5 \times 6 = 15 \text{ inches}$$The enlarged photo also covers \(2.5^2 = 6.25\) times the original area.
FAQ
What does a scale factor less than 1 mean? It is a reduction — the new figure is smaller than the original. For example, \(k = 0.5\) halves every length.
How do areas change with scale factor? Area scales by \(k^2\), so a scale factor of 3 makes the area 9 times larger.
Can the scale factor be negative? In pure geometry of similar figures it is taken as positive; negative values appear only in coordinate transformations involving reflection, which this calculator does not model.
