a scale factor
A scale factor is the single number that tells you how much bigger or smaller one similar figure is than another. If you blow a picture up to twice its size, the scale factor is 2; if you shrink a map down so 5 km becomes 1 cm, the scale factor from reality to map is 1/500000. It is simply the common ratio of corresponding lengths between two similar figures.
Given two similar figures, the scale factor k is found by dividing any length in the new figure by the corresponding length in the original: k = (new length)/(old length). Every corresponding pair gives the same k — that consistency is precisely what makes the figures similar. A scale factor greater than 1 is an enlargement, between 0 and 1 a reduction, and exactly 1 means the figures are congruent. To go back the other way you simply use the reciprocal 1/k.
The crucial warning is that the scale factor acts differently on length, area, and volume. Lengths multiply by k, but areas multiply by k^2 and volumes by k^3. Double every edge of a box and its surface area quadruples while its volume becomes eight times larger. Forgetting this is one of the most common mistakes in the subject: a map scale of 1 : 100 does not mean a region's area on the map is 1/100 of the real area — it is 1/10000. This square-cube behaviour governs scale models, dosing by body size, and why large animals cannot have the proportions of small ones.
A model train is built at scale 1 : 87. A real carriage 8.7 m long becomes 8700 cm / 87 = 100 cm — wait, 870 cm / 87 = 10 cm long on the model. Its painted side area shrinks by 87^2 = 7569, and its weight (volume) by 87^3.
One linear scale factor; length by k, area by k^2, volume by k^3.
Unless stated otherwise, 'scale factor' means the linear (length) factor. Areas and volumes scale by its square and cube — never apply the same k to an area as you would to a length.