A scale factor is the multiplier between a drawing or model and the real thing. A 1:87 model is one eighty-seventh of full size, so a 12 cm model represents 1,044 cm in reality. The arithmetic is a single multiplication, which is why scales are written as ratios rather than as instructions.
What surprises people is that area and volume do not scale by the same factor. Doubling every length quadruples the area and multiplies the volume by eight, because area involves two dimensions and volume three. A 1:87 model has roughly one seven-thousand-five-hundredth of the surface area and one six-hundred-and-fifty-eight-thousandth of the volume, which is why scale models weigh so little.
The formula
S- The scale factor, the second number in a 1:S ratio
Model- The measurement on the drawing or model
Real- The full-size measurement
How it works, step by step
- Enter the scale denominator — 87 for a 1:87 model.
- Enter the measurement you have on the model.
- The gauge shows the equivalent full-size length.
- The area and volume factors below show why models are so much lighter than they look.
Worked examples
A 12 cm model at 1:87
12 × 87 = 1,044 cm, which is 10.44 m full size — about the length of a bus. HO scale is 1:87 precisely because it makes standard rolling stock this size.
Why models are so light
At 1:87 the volume factor is 87 cubed, or 658,503. A vehicle weighing 12 tonnes would scale to 18.22 grams if it were made of the same material.
How to read your score
Frequently asked questions
What does a scale of 1:50 actually mean?
Every length on the drawing represents fifty of the same units in reality. One centimetre on the page is fifty centimetres, or half a metre, on the building.
Why does area scale by the square of the factor?
Because area is a length multiplied by a length, and both get scaled. Halving every dimension leaves a quarter of the area, not half.
How do I convert a map scale into distance?
Multiply the measured distance by the scale denominator. Four centimetres on a 1:25000 map is 100,000 cm, which is one kilometre.
Is a larger scale number a bigger or smaller model?
Smaller. 1:24 is a large model and 1:500 is a tiny one, because the denominator counts how many times reality is bigger than the model.
Length, area and volume at common scales
| Scale 1:S | Length factor | Area factor | Volume factor | 12 cm becomes |
|---|---|---|---|---|
| 10 | 10 | 100 | 1,000 | 1.20 m |
| 24 | 24 | 576 | 13,824 | 2.88 m |
| 43 | 43 | 1,849 | 79,507 | 5.16 m |
| 50 | 50 | 2,500 | 125,000 | 6.00 m |
| 72 | 72 | 5,184 | 373,248 | 8.64 m |
| 87 | 87 | 7,569 | 658,503 | 10.44 m |
| 100 | 100 | 10,000 | 1e+06 | 12.00 m |
| 200 | 200 | 40,000 | 8e+06 | 24.00 m |
| 500 | 500 | 250,000 | 1.25e+08 | 60.00 m |
Area grows with the square of the scale factor and volume with its cube.