A linear scale factor is the number you multiply every length by to go from one similar shape to another: scale factor = new length ÷ old length. It works the same for triangles, rectangles, scale models and maps.
It builds on corresponding sides, so check your pairs first.
Why multiply and not add?
Similar shapes keep their proportions, so every length is scaled by the same factor. If one side doubles, every side doubles. Adding the same amount to each side would change the shape.
Think of photocopying a page at 150%. A 10 cm line becomes 15 cm, and a 4 cm line becomes 6 cm. The lines do not grow by the same number of centimetres, but they grow by the same factor, 1.5.
The method
- Identify the pairs of corresponding sides.
- Find one known pair, where both lengths are given.
- Work out the scale factor, new ÷ old, and decide whether it should be above or below 1.
- Multiply the length you want to scale by that factor, or divide if you are going the other way.
- Check that the answer is larger if you enlarged, smaller if you reduced.
Worked example
Two triangles are similar. The smaller has sides 4.5 cm, 6 cm and 7.5 cm.
The longest side of the larger triangle is 12.5 cm. Find the other two sides of the larger triangle.
Step 1, the known pair. Longest matches longest: 7.5 cm and 12.5 cm.
Step 2, scale factor. 12.5 ÷ 7.5 = 5/3 (about 1.667).
Step 3, the other sides. 4.5 × 5/3 = 7.5 cm, and 6 × 5/3 = 10 cm.
Step 4, check. The sides are 7.5, 10 and 12.5, which are in increasing order like 4.5, 6 and 7.5. Also 7.5 × 5/3 = 12.5, so the factor is consistent.
A map example. A map has scale 1 : 50 000. A road measures 3.2 cm on the map. Real length = 3.2 × 50 000 = 160 000 cm. Since 100 cm = 1 m, that is 1600 m, which is 1.6 km.
The mistake to watch for
A common slip is to find the difference between two corresponding sides and add it to every side.
Mistaken working: 12.5 − 7.5 = 5, so add 5 to each side: 4.5 + 5 = 9.5 and 6 + 5 = 11.
The new sides 9.5, 11 and 12.5 do not keep the original proportions, so the triangles would not be similar.
The correction is to divide, not subtract. The scale factor is a ratio, 12.5 ÷ 7.5, and each side is multiplied by it. A useful test: the ratio of any two sides must stay the same in both triangles.
Check yourself
Try these, then open each answer.
1. Rectangle A measures 8 cm by 5 cm. Rectangle B is similar, with its longer side 20 cm. Find its shorter side.
Show answer
Scale factor = 20 ÷ 8 = 2.5. The shorter side is 5 × 2.5 = 12.5 cm. Check: 8 × 2.5 = 20, so the same factor works for both sides.
2. A model car is built to scale 1 : 40. The model is 9 cm long. How long is the real car in metres?
Show answer
Real length = 9 × 40 = 360 cm. Since 100 cm = 1 m, that is 3.6 m.
3. A shape is reduced so that a side of 18 cm becomes 12 cm. What does a side of 7.5 cm become?
Show answer
Scale factor = 12 ÷ 18 = 2/3. Then 7.5 × 2/3 = 5 cm. The factor is below 1 and the answer is smaller, as it should be for a reduction.
Where this leads next
Lengths are only one kind of measurement. The next step is to see what the same factor does to area, and later to volume. The map scale, contour and gradient practice tool and the non-calculator working trainer are useful for extra repetitions.
Some students handle the calculation but hesitate over which shape is “new” and which is “old”. A teacher in online one-to-one Mathematics tuition can listen to how you choose, and fix that decision point.