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Scale Factor Calculator
Find the scale factor between two similar figures from a pair of matching lengths, areas or volumes, then apply it to every side. The calculator shows whether it is an enlargement or a reduction, the area factor k² and the volume factor k³.
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Find the scale factor from
A side of the original (pre-image) figure
The matching side of the new (image) figure
e.g. 2, 0.25, 3/4 or a ratio original:new like 1:4
Apply the scale factor (optional)
Separate with commas or spaces; a sum (perimeter) is included
units²
units³
Enter your values to see the result.
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How to use the Scale Factor Calculator
- Pick what you know: two lengths, a scale factor, two areas or two volumes.
- Enter the values from the original figure and the new figure.
- List other lengths of the original figure (optional) to scale them all at once.
- Add an area or volume (optional) to see it scaled by k² or k³.
- Press Calculate and read the scale factor, the factors for area and volume and the steps.
Scale factor formulas
- From lengths: k = new length ÷ original length
- Scaled length: new length = k × original length (perimeter too)
- Scaled area: new area = k² × original area, so k = √(new area ÷ original area)
- Scaled volume: new volume = k³ × original volume, so k = ∛(new volume ÷ original volume)
- Reverse: going from the new figure back to the original uses 1/k
The figures must be similar: same shape, equal angles and every pair of corresponding sides in the same ratio. If different side pairs give different values of k, the shapes are not similar.
How k changes length, area and volume
| Scale factor k | Type | Lengths | Areas (k²) | Volumes (k³) |
|---|---|---|---|---|
| 1/4 | Reduction | × 0.25 | × 0.0625 | × 0.015625 |
| 1/2 | Reduction | × 0.5 | × 0.25 | × 0.125 |
| 3/4 | Reduction | × 0.75 | × 0.5625 | × 0.421875 |
| 1 | Congruent | × 1 | × 1 | × 1 |
| 1.5 | Enlargement | × 1.5 | × 2.25 | × 3.375 |
| 2 | Enlargement | × 2 | × 4 | × 8 |
| 3 | Enlargement | × 3 | × 9 | × 27 |
| 10 | Enlargement | × 10 | × 100 | × 1,000 |
Tips
- Always divide new by original. Dividing the other way gives 1/k, which describes the reverse transformation.
- Match corresponding sides carefully: the shortest side goes with the shortest side, the side opposite the same angle with its partner.
- Keep units the same for both lengths; k itself has no units.
- A common mistake is doubling the area when the lengths double. Doubling the sides makes the area four times as big and the volume eight times as big.
- In a dilation on a coordinate grid, multiply each coordinate by k when the center of dilation is the origin.
Frequently asked questions
How do you find the scale factor?
Divide a length on the new figure by the matching length on the original: k = new ÷ original. If a side of 4 cm becomes 10 cm, k = 10 ÷ 4 = 2.5. The two figures must be similar, so every pair of matching sides gives the same k.
What does a scale factor less than 1 mean?
A scale factor between 0 and 1 is a reduction: the new figure is smaller. k = 0.25 makes every length a quarter of the original. A scale factor above 1 is an enlargement and k = 1 means the figures are congruent.
How does the scale factor affect area?
Areas change by the square of the scale factor. If k = 3, every area is 3² = 9 times larger. To go back from areas to k, take the square root of new area ÷ original area.
How does the scale factor affect volume?
Volumes change by the cube of the scale factor. Doubling every length (k = 2) makes the volume 2³ = 8 times larger. To find k from two volumes, take the cube root of their ratio.
Is the scale factor the same as a ratio like 1:4?
They describe the same thing. A ratio original : new of 1 : 4 means k = 4 (enlargement); 4 : 1 means k = 1/4 (reduction). Map and model scales such as 1:24,000 compare the drawing to real life, so the factor from real to map is 1/24,000.
Does the perimeter scale by k too?
Yes. Perimeter, circumference and any other length scale by k, while angles stay exactly the same in similar figures.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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