# What is the scale factor?

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.

- Page: https://www.acalculator.org/math/scale-factor-calculator
- JSON spec: https://www.acalculator.org/math/scale-factor-calculator.json
- Version: 689d25cf69d1

## Default answer

Example with the default inputs (Original length 4, New length 10): From 4 to 10, the scale factor is 2.5 (2.5 : 1).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Original length | A length on the original figure, in any unit. |
| b | New length | The matching length on the new (scaled) figure, in the same unit. |
| k | Scale factor | How many times larger the new figure is: new length ÷ original length. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| a | Original length | A length on the original figure, in any unit. |
| b | New length | The matching length on the new (scaled) figure, in the same unit. |
| k | Scale factor | New length ÷ original length. |
| fraction | Scale factor as a fraction | The factor as a fraction in lowest terms, from the two lengths as typed (shown when its bottom is at most 1,000,000). |
| ratio | Scale ratio | The scale as new : original, written 1 : n for a reduction and k : 1 for an enlargement. |
| area | Area scale factor | How many times larger every area is: k². |
| volume | Volume scale factor | How many times larger every volume is: k³. |
| kind | Type of scaling | An enlargement when k > 1, a reduction when k < 1, the same size when k = 1. |

## Method

k = b ÷ a; b = k × a; a = b ÷ k; area factor = k²; volume factor = k³.

## Assumptions

- 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

1. a = 4, b = 10 gives k = 2.5, area = 6.25, volume = 15.625, ratio = 2.5 : 1, kind = 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).
2. a = 1.3, k = 270 gives b = 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).
3. a = 9, b = 3 gives k = 0.333333, ratio = 1 : 3, kind = Reduction, volume = 0.037037.

## FAQ

### 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.

## Sources

- OpenStax, Elementary Algebra 2e, section 8.7, Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio; Example 8.79, map scale), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications