Scale Factor Calculator
A scale factor is the number you multiply every length by to turn one shape into a similar one: k = new length ÷ original length. This scale factor calculator finds k from two lengths or applies k to a length, says whether the result is an enlargement or a reduction, and gives the matching area factor (k²) and volume factor (k³). Below are maps, models, similar shapes and the common mistakes.
1/4.How to use the scale factor calculator
There are two modes, chosen with the buttons above the inputs.
- Find the scale factor (the default). Enter an original length and the new length. With the defaults, original = 4 and new = 10, the calculator returns k = 2.5, labels it an enlargement, and also writes it as the ratio 5:2 and as 250%.
- Apply a scale factor. Enter an original length and k. With original = 4 and k = 2.5 the new length is 10. You can type k as a fraction, such as
1/4, which is how map and model scales are usually written.
In both modes the result panel also shows the area factor and the volume factor. For k = 2.5 these are 2.5² = 6.25 and 2.5³ = 15.625: the bigger shape has 6.25 times the area and, if it is a solid, 15.625 times the volume. When k = 1 the calculator reports that the shapes are congruent, meaning the same size.
Both lengths must be in the same unit. If one is in inches and the other in feet, convert first, or the factor will be off by 12.
The scale factor formula
Rearranged, the same formula applies a scale factor (new = original × k) or works backward to the original (original = new ÷ k). The direction matters. Going from the original to the copy gives k; going the other way gives 1 ÷ k. From 4 to 10 the factor is 2.5, while from 10 back to 4 it is 1 ÷ 2.5 = 0.4.
| Value of k | What it means | Example |
|---|---|---|
| k > 1 | Enlargement | 4 → 10, k = 2.5 |
| k = 1 | Congruent (same size) | 4 → 4, k = 1 |
| 0 < k < 1 | Reduction | 12 → 3, k = 0.25 |
A scale factor can be written three ways, and the calculator shows all three. As a decimal, 2.5. As a ratio of new to original, 10:4, which simplifies to 5:2 in the same way the ratio calculator reduces any ratio. As a percentage, 250%, which means the copy is 250% of the original size, or 150% larger. A reduction to 0.8 is 80% of the original, a 20% decrease, the same reading the percentage change calculator gives for 4 → 3.2.
Worked examples
Example: the default enlargement, 4 to 10
A side 4 units long becomes 10 units long.
Example: a reduction, 12 to 3
A 12 cm line is drawn 3 cm long.
Every length is a quarter of the original, every area a sixteenth. The fraction 3/12 and the fraction 1/4 are the same number, which the equivalent fractions guide explains. If you prefer fractions to decimals, the decimal to fraction calculator turns 0.0625 back into 1/16.
Example: finding a missing side in similar shapes
Two rectangles are similar. The small one is 6 by 8. The large one has a short side of 9. How long is its long side?
The larger rectangle is 9 by 12. Matching sides is the whole job: dividing 9 by 8 would pair a short side with a long side and give the wrong factor.
The missing-side question is a proportion in disguise, 6/8 = 9/x. When the setup is a word problem with rates rather than shapes, the proportion calculator is the better fit.
Scale factor in similar shapes
Two shapes are similar when one is an exact enlargement or reduction of the other: all angles match and every pair of corresponding sides has the same ratio. That shared ratio is the scale factor.
If even one pair gives a different ratio, the shapes are not similar, however alike they look. A 4 by 6 rectangle and a 6 by 8 rectangle fail the test: 6 ÷ 4 = 1.5 but 8 ÷ 6 ≈ 1.333.
In coordinate geometry, a dilation centered on the origin applies k to every coordinate. The point (2, 3) under a dilation with k = 3 moves to (6, 9), and every distance in the figure is three times as long.
Area and volume scale factors
Lengths scale by k, but area has two dimensions and volume has three, so they scale by k multiplied by itself that many times.
| Length factor k | Area factor k² | Volume factor k³ |
|---|---|---|
| 1/4 = 0.25 | 0.0625 | 0.015625 |
| 1/2 = 0.5 | 0.25 | 0.125 |
| 0.8 | 0.64 | 0.512 |
| 1.5 | 2.25 | 3.375 |
| 2 | 4 | 8 |
| 2.5 | 6.25 | 15.625 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 10 | 100 | 1,000 |
When k is a whole number, the area factors in the middle column are the perfect squares 4, 9, 16 and 100, which makes them quick to recognize when you work backward.
Example: enlarging a photo
A 4 × 6 inch print is enlarged to 8 × 12 inches.
Doubling the sides quadruples the paper. A print twice as wide needs four times the ink, not twice.
Example: a pizza comparison
A 16-inch pizza against a 12-inch pizza.
The larger pizza is only a third wider but has about 78% more pizza. That is why price per square inch, not price per inch of diameter, is the fair comparison, and the unit price calculator can do that division once you have both areas.
Working back from area or volume
If you know how areas compare, take the square root to get k. Two similar rugs cover 20 and 45 square feet: 45 ÷ 20 = 2.25, and √2.25 = 1.5, so every length on the bigger rug is 1.5 times as long. For volumes, take the cube root: a volume ratio of 8 means k = 2. The square root calculator handles both square and cube roots.
The same rule explains why a cube box with sides twice as long uses 4 times the cardboard (surface area, k² = 4) and holds 8 times as much (volume, k³ = 8), with 3 cm sides going from 27 cm³ to 216 cm³.
Scale factors on maps, models and blueprints
Maps, models and plans state their scale as a ratio 1:n, where 1 unit on the drawing stands for n of the same unit in real life. The scale factor from real to drawing is 1/n; from drawing to real it is n.
Example: a 1:50,000 map
Two towns are 4 cm apart on the map.
Example: a 1:18 model car
The real car is 4.5 m long.
In the calculator’s apply mode, enter 4.5 as the original and type 1/18 as k.
Example: a blueprint at 1/4 inch = 1 foot
Convert to one unit first. One foot is 12 inches, so a quarter inch for 12 inches is 1 for 48: the scale is 1:48.
Map scales written in words need the same unit conversion. “One inch to 20 miles” is 1 inch to 20 × 63,360 = 1,267,200 inches, so the ratio scale is 1:1,267,200. Large scale denominators like that are easier to handle in scientific notation.
Common mistakes
- Dividing the wrong way round. The scale factor from original to copy is new ÷ original. Dividing original ÷ new gives the reciprocal, 0.4 instead of 2.5.
- Pairing sides that do not correspond. Short side to short side, long side to long side. Rotated or flipped shapes make this easy to get wrong.
- Using k for area or volume. Doubling lengths doubles nothing but length. Area goes up 4 times and volume 8 times.
- Mixing units. A 6-inch model of a 6-foot object has a scale factor of 1/12, not 1.
- Adding instead of multiplying. Going from 4 to 10 is not “add 6 to every side.” A side of 8 becomes 20, not 14.
- Reading a percentage as the increase. k = 250% means 2.5 times the size, which is 150% larger, not 250% larger.
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 4 becomes 10, then k = 10 ÷ 4 = 2.5. A result above 1 is an enlargement, below 1 is a reduction, and exactly 1 means the shapes are the same size. Always pair corresponding sides.
What is the scale factor from 12 to 3?
The scale factor is 0.25, or 1/4. Divide the new length by the original: 3 ÷ 12 = 0.25. It is a reduction, so every length is a quarter of the original, every area is (1/4)² = 1/16 of the original, and any volume is (1/4)³ = 1/64 of it.
How does a scale factor affect area?
Area changes by the square of the scale factor. With k = 2, a rectangle’s area becomes 2² = 4 times as large: a 4 × 6 inch photo with 24 square inches becomes 8 × 12 with 96 square inches. With k = 2.5 the area factor is 6.25. For volume, cube k instead.
Can a scale factor be a fraction or less than 1?
Yes. Any scale factor between 0 and 1 is a reduction, and it is often written as a fraction. A factor of 1/4 makes every length a quarter as long, which is how a 12 cm line becomes 3 cm. Model and map scales such as 1:18 or 1:50,000 are fractional scale factors, 1/18 and 1/50,000.
How do you find the scale factor of two similar triangles?
Divide each side of one triangle by the matching side of the other. For sides 3, 4, 5 and 7.5, 10, 12.5, every division gives 2.5, so the scale factor is 2.5 and the triangles are similar. If the three divisions do not all give the same number, the triangles are not similar.
What does a map scale of 1:50,000 mean?
One unit on the map stands for 50,000 of the same unit on the ground. Four centimeters on the map is 4 × 50,000 = 200,000 cm, which is 2 km. The scale factor from the ground to the map is 1/50,000. The ratio works in any unit, as long as both sides use the same one.
How do you find the scale factor from two areas?
Divide the areas, then take the square root. Similar rugs of 20 and 45 square feet have an area ratio of 45 ÷ 20 = 2.25, and √2.25 = 1.5, so the length scale factor is 1.5. For two volumes, divide and take the cube root instead: a volume ratio of 27 gives a scale factor of 3.
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