About the Scale Factor Calculator
Two shapes are similar when one is an exact enlargement or reduction of the other. The scale factor k is the ratio of matching lengths: new length ÷ original length. This scale factor calculator finds k from a pair of corresponding sides, applies a known k to get a scaled length, or works backwards from a scaled length to the original.
Because area grows with the square of the scale factor and volume with the cube, it also scales an area and a volume you enter. That is the part people most often get wrong: doubling every dimension of a box makes it hold eight times as much, not twice as much. The tool is useful for geometry dilations, scale drawings and models, resizing recipes for round or square pans, enlarging photos and plans, and model railways or miniatures.
A scale factor above 1 is an enlargement, between 0 and 1 a reduction, and exactly 1 means the shapes are congruent. Lengths can be in any unit as long as both are in the same one.
With the default inputs, the scale factor (k) is 2.5. Change any value above to recalculate instantly.
How to use the scale factor calculator
- 1Choose whether to find the scale factor, scale a length, or recover the original.
- 2Enter the lengths (same unit) and/or the scale factor.
- 3Optionally enter an original area and volume to scale them too.
- 4Read the scale factor, the scaled length, and the area and volume multipliers.
Formula and method
For similar figures every pair of matching lengths has the same ratio, the scale factor k. Scaling a length multiplies it by k; going back divides by k.
Area is a product of two lengths, so each is multiplied by k and the area is multiplied by k × k = k². Volume is a product of three lengths, so it is multiplied by k³. A model built at k = 1/10 therefore has 1/100 of the surface area and 1/1,000 of the volume of the real object. The percentage change in length is (k − 1) × 100.
- k
- Linear scale factor
- k²
- Area scale factor
- k³
- Volume scale factor
Worked examples
Side 4 becomes side 10
k = 10 ÷ 4 = 2.5, an enlargement. An area of 12 becomes 12 × 6.25 = 75 and a volume of 8 becomes 8 × 15.625 = 125. Each length grows by 150%.
Halve a shape (k = 0.5)
Halving every length takes 9 to 4.5. Area shrinks to a quarter (36 → 9) and volume to an eighth (216 → 27).
1:50 scale drawing — real size of a 6 cm wall
On a 1:50 plan every real length is 50 times the drawn one, so 6 cm on paper is 300 cm (3 m) in reality, and every area on the plan represents 2,500 times as much floor.
Model is 12 cm at k = 1/24 — how big is the original?
Dividing by the scale factor undoes the scaling: 12 ÷ (1/24) = 12 × 24 = 288 cm for the real object.
Frequently asked questions
How do you find the scale factor?+
Divide a length on the new shape by the matching length on the original: k = new ÷ original. If a side of 5 cm becomes 15 cm, the scale factor is 3.
How does scale factor affect area?+
Area is multiplied by the square of the scale factor. Scaling a shape by 3 multiplies its area by 9; scaling by 0.5 leaves a quarter of the area.
How does scale factor affect volume?+
Volume is multiplied by the cube of the scale factor. Doubling every dimension of a box multiplies its volume by 2³ = 8.
What does a scale factor less than 1 mean?+
A scale factor between 0 and 1 is a reduction: the new shape is smaller. For example, k = 0.25 makes every length a quarter of the original.
How do I use a scale like 1:100?+
A 1:100 scale means 1 unit on the drawing equals 100 units in reality. Multiply drawing lengths by 100 to get real lengths, or divide real lengths by 100 to draw them.