What is the scale factor?
Type a length on the original figure and the matching length on the new one. The scale factor calculator divides them and shows how areas and volumes change; fill the factor instead to find a length.
- Scale factor
- 2.5
From 4 to 10, the scale factor is 2.5 (2.5 : 1).
- Scale factor as a fraction
- 2 1/2
- Scale ratio
- 2.5 : 1
- Area scale factor
- 6.25
- Volume scale factor
- 15.625
- Type of scaling
- Enlargement
Scale factor: 2.5. From 4 to 10, the scale factor is 2.5 (2.5 : 1).
How to calculate
Computes the scale factor between two similar figures from an original and a new length (k = new ÷ original), or a length from the factor, with the area factor k², the volume factor k³ and the scale ratio.
Example with the default inputs (Original length 4, New length 10): From 4 to 10, the scale factor is 2.5 (2.5 : 1).
Formula: k = b ÷ a; b = k × a; a = b ÷ k; area factor = k²; volume factor = k³.
- The two figures are similar: every length is scaled by the same factor.
- Both lengths are in the same unit, and are more than 0.
Worked examples
Each example is checked against the calculator on every build.
- Original length 4, New length 10 gives Scale factor 2.5, Area scale factor 6.25, Volume scale factor 15.625, Scale ratio 2.5 : 1, Type of scaling Enlargement.Source: OpenStax Elementary Algebra 2e, section 8.7, Solve Proportion and Similar Figure Applications: corresponding sides of similar figures are in the same ratio (https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications, retrieved 2026-10-01)
- Original length 1.3, Scale factor 270 gives New length 351.Source: OpenStax Elementary Algebra 2e, section 8.7, Example 8.79: 1 inch on the map is 270 miles, 1.3 inches is 351 miles (https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications)
- Original length 9, New length 3 gives Scale factor 0.333333, Scale ratio 1 : 3, Type of scaling Reduction, Volume scale factor 0.037037.
How it works
Fill any two of original length a, new length b and scale factor k, and leave one empty:
- k = b ÷ a
- b = k × a
- a = b ÷ k
Then the calculator shows:
- Scale factor k, to 6 significant figures.
- Scale factor as a fraction: b ÷ a as a fraction in lowest terms, from each length as typed (its shortest decimal). It is written as a mixed number when it is above 1 (10 and 4 give 2 1/2) and is shown only when the bottom of the fraction is at most 1,000,000; 9 and 3 give 1/3.
- Scale ratio: "k : 1" when k is 1 or more (2.5 : 1), and "1 : n" with n = 1 ÷ k when k is less than 1 (1 : 3). The number is written to at most 6 significant figures, rounded half up from its exact value (359.9435 : 1 shows 359.944 : 1).
- Area scale factor k² and volume scale factor k³, worked from the exact factor (b ÷ a from the typed lengths when that is the factor, else k as typed), to 10 significant figures.
- Type of scaling: "Enlargement" when k > 1, "Reduction" when k < 1, "Same size" when k = 1.
Values are rounded half up for display only.
Rules
- Both lengths are more than 0, in the same unit.
- The scale factor is from 10⁻¹⁰⁰ to 10¹⁰⁰, so its cube is always a number; a factor solved outside this range gives no answer: "The scale factor is outside 10⁻¹⁰⁰ to 10¹⁰⁰."
- A length solved beyond about 1.8 × 10³⁰⁸ (too large to hold as a number) gives no answer.
Worked examples by hand
4 to 10. k = 10 ÷ 4 = 2.5, an enlargement with the ratio 2.5 : 1. Area factor 2.5² = 6.25; volume factor 2.5³ = 15.625.
The map. 1 inch is 270 miles, so k = 270. A distance of 1.3 inches is b = 270 × 1.3 = 351 miles.
9 to 3. k = 3 ÷ 9 = 1/3 = 0.333…, a reduction with the ratio 1 : 3. Volume factor (1/3)³ = 1/27 = 0.037037….
Other questions people ask
How do I find the scale factor?
Divide a length on the new figure by the matching length on the original: k = new ÷ original. A side that grows from 4 cm to 10 cm has a scale factor of 10 ÷ 4 = 2.5.
What does a scale factor less than 1 mean?
The new figure is smaller: a reduction. A side that shrinks from 9 to 3 has a scale factor of 1/3, written as the ratio 1 : 3.
How does the scale factor change the area?
Areas scale by the square of the factor, k², because an area multiplies two lengths that each grow by k. With k = 2.5, every area is 6.25 times larger.
How does the scale factor change the volume?
Volumes scale by the cube, k³, because a volume multiplies three lengths. With k = 2.5, every volume is 15.625 times larger.
How do I use a map scale?
Multiply the map distance by the scale factor. If 1 inch on a map is 270 miles, then 1.3 inches is 1.3 × 270 = 351 miles. Type 1.3 as the original length and 270 as the scale factor.
Do the lengths need the same unit?
Yes. The scale factor has no unit, so convert both lengths to the same unit first. A 1 : 24 model of a 4.5 m car is 4.5 ÷ 24 = 0.1875 m long.