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Width & Height Ratio Calculator
Get the aspect ratio from a width and height, or resize to a locked ratio such as 16:9 by entering just one dimension.
Open → Width & Height Ratio CalculatorScale Factor Calculator
Find the scale factor between a model and the real thing, or apply a known scale to any dimension. Both modes convert units for you and report the area and volume factors, which are the parts people most often get wrong.
Formula k = new ÷ original
Common scales
Try an example
Result
| Quantity | Scales by | Factor | Meaning |
|---|
Method
One division, then square and cube
Finding the factor is easy. Remembering that area and volume do not scale the same way is the part that needs attention.
Match a pair of corresponding lengths
They must measure the same feature — the model's length against the real length, not the model's length against the real height. Any corresponding pair gives the same factor, so pick whichever you know most accurately.
Put both in the same unit
8 cm and 4 m cannot be divided as they stand. Convert one:
4 m = 400 cm. This step is where most wrong answers originate, and the
error is usually a factor of 10, 100 or 1000 — large enough to spot if you sanity
check the result.
Divide new by original
k = 8 ÷ 400 = 0.02. Under 1 means a reduction, over 1 an enlargement.
Getting the division the wrong way round gives 50 instead of 0.02 — check whether
the answer matches the direction you expect.
Express it as a ratio
Take the reciprocal for the standard form: 1 ÷ 0.02 = 50, so the scale
is 1:50. Model and map conventions always use this form because
"one to fifty" is easier to hold in mind than "nought point nought two".
Square for area, cube for volume
k² = 0.0004 and k³ = 0.000008. Written as ratios that is
1:2500 for area and 1:125,000 for volume. The model has one two-thousand-five-hundredth
of the surface to paint and one hundred-and-twenty-five-thousandth of the volume to
fill.
k = new ÷ original
Linear scale factor. One number that multiplies every length. Same units on both sides, always.
area × k² volume × k³
The derived factors. Area has two dimensions of length, so it picks up two factors of k. Volume has three. This is not an approximation — it is exact for any shape.
The reduce-then-enlarge trap
Shrinking a drawing to 80% and then enlarging it by 20% does not restore it:
0.8 × 1.2 = 0.96, leaving it 4% small. To undo a scaling you need the
reciprocal — 1 ÷ 0.8 = 1.25, so 125%.
Worked examples
Models, maps and drawings
Model car
A model of a 4 m car is 8 cm long. What is the scale?
- Same units
- 4 m = 400 cm
- Divide
- k = 8 ÷ 400 = 0.02
- As a ratio
- 1 ÷ 0.02 = 50, so 1:50
- Area factor
- 0.02² = 0.0004 (1:2500)
- Volume factor
- 0.02³ = 0.000008 (1:125,000)
Scale 1:50, factor 0.02
Map distance
On a 1:25,000 map, two points are 6 cm apart. How far is that on the ground?
- Scale means
- 1 cm map = 25,000 cm real
- Multiply
- 6 × 25,000 = 150,000 cm
- Convert
- 150,000 cm = 1500 m
- Or
- 1.5 km
1.5 km
Floor plan
A room is 5.4 m wide. How wide is it on a 1:100 plan?
- Factor
- k = 1/100 = 0.01
- Multiply
- 5.4 × 0.01 = 0.054 m
- Convert
- 0.054 m = 5.4 cm
- Floor area
- scales by 0.01² = 1:10,000
5.4 cm on the plan
Paint estimate
A 1:20 model needs 15 ml of paint. How much for the real object?
- Paint covers
- area, not length
- Area factor
- 20² = 400
- Multiply
- 15 × 400 = 6000 ml
- Which is
- 6 litres, not 300 ml
6 litres — 400×, not 20×
Reference
Standard scales and what they mean
| Scale | Factor | Used for | 1 unit on paper is… | Area factor |
|---|---|---|---|---|
| 1:1 | 1 | Full-size detail drawings | 1 unit | 1 |
| 2:1 | 2 | Watch and electronics parts | 0.5 unit | 4 |
| 1:10 | 0.1 | Furniture, machine details | 10 units | 1/100 |
| 1:20 | 0.05 | Room and joinery details | 20 units | 1/400 |
| 1:24 | 0.0417 | Die-cast model cars | 24 units | 1/576 |
| 1:50 | 0.02 | Architectural plans | 50 units | 1/2500 |
| 1:87 | 0.0115 | HO gauge model railways | 87 units | 1/7569 |
| 1:100 | 0.01 | Floor plans, site layouts | 100 units | 1/10,000 |
| 1:1250 | 0.0008 | Urban site plans | 1250 units | 1/1,562,500 |
| 1:25,000 | 0.00004 | Walking and hiking maps | 25,000 units | 1/625,000,000 |
| 1:50,000 | 0.00002 | Road and topographic maps | 50,000 units | 1/2.5 × 10⁹ |
Background
Why area and volume misbehave
The square and cube laws are the least intuitive part of scaling, and the source of most real-world errors. The reason is structural: an area is a length multiplied by a length, so if every length is multiplied by k, the area picks up k twice. A volume is three lengths multiplied together, so it picks up k three times. Nothing about the shape matters — the rule holds for a cube, a sphere, a car or a cathedral.
The consequences are dramatic once k gets far from 1. Doubling a pizza's diameter gives you four times the pizza. A 1:50 model has one two-thousand-five-hundredth of the surface area — which is why a tiny pot of paint covers an entire model kit. Halving the linear dimensions of a shipping box cuts its capacity to one eighth, not one half.
The reason large animals are shaped differently
Weight is proportional to volume, so it grows as k³. But the strength of a
bone depends on its cross-sectional area, which grows only as k². Scale an
animal up by a factor of 10 and it becomes 1000 times heavier while its legs become only
100 times stronger — the load per unit of bone has increased tenfold. This is why an
elephant has thick, pillar-like legs while a gazelle can have thin ones, and why a
hypothetical scaled-up insect could not stand. Galileo worked this out in 1638, and it is
the same square–cube arithmetic this calculator reports.
Scale factors and similar shapes
Two shapes are geometrically similar when one is a scaled copy of the other: all corresponding angles equal, all corresponding lengths in the same ratio. That common ratio is the scale factor. It follows that a single measured pair is enough to determine every other length, which is what makes similar triangles such a powerful surveying tool — measure one accessible length and one shadow, and you can find the height of a building you cannot reach.
Related tools
For the ratio side of the same problem, the simplify ratio calculator reduces a measured pair like 8:400 to 1:50, and the equivalent ratio calculator generates the series of matching dimensions. If you are resizing an image or a screen layout, the width and height ratio calculator handles aspect ratios specifically. And where scaling one quantity drives another linearly, the direct proportion calculator is the same maths in a different frame.
Questions
Scale factor — questions
Including the area and volume rules, which is where nearly every real mistake happens.
What is a scale factor?
A scale factor is the number every length is multiplied by when a shape is enlarged or reduced. It is found by dividing a new length by the corresponding original length:
k = new ÷ original. A scale factor greater than 1 is an enlargement, between 0 and 1 is a reduction, and exactly 1 leaves the shape unchanged. Scale factors are often written as a ratio instead — a factor of 0.02 is the same as a scale of 1:50.How do you calculate a scale factor?
Pick one length that you know in both the model and the real object, then divide. If a model car is 8 cm long and the real car is 4 m long, convert to the same unit first — 4 m is 400 cm — and divide:
8 ÷ 400 = 0.02, which is a scale of 1:50. Forgetting to convert units is the single most common error, which is why this calculator has a unit selector on each length.What is the difference between scale factor and scale ratio?
They express the same thing differently. The scale factor is a single multiplier (0.02); the scale ratio compares model to real as two numbers (1:50). Convert between them by taking the reciprocal: a factor of
1/50 is a ratio of 1:50, and a factor of 3 is a ratio of 3:1. Architectural and map work almost always uses the ratio form because it reads more naturally.How does scale factor affect area and volume?
Area scales by the square of the linear factor, and volume by the cube. Double every length and the area quadruples while the volume becomes eight times bigger. This catches people out constantly: a 1:50 model has 1/2500 of the surface area and 1/125,000 of the volume of the real thing, not 1/50. It is why a scale model needs vastly less paint than you would guess, and why large animals cannot simply be scaled-up small ones.
How do I read a map scale like 1:25,000?
One unit on the map represents 25,000 of the same unit on the ground. So 1 cm on the map is 25,000 cm — that is 250 m, or 0.25 km. To convert a measured map distance to a real one, multiply by the second number; to go the other way, divide. Entering the scale into the "apply a scale" mode here does the arithmetic and the unit conversion together.
What is a scale factor of 1:1?
Full size — the drawing or model is exactly as large as the real object. Engineering drawings often specify 1:1 for detail views so that a part can be checked against the paper directly. A scale of 2:1 means the drawing is twice life size, which is normal for small components like watch parts or electronics.
Can a scale factor be negative?
In transformation geometry, yes. A negative scale factor enlarges the shape and also rotates it 180° about the centre of enlargement, so the image ends up on the opposite side and upside down. A factor of −2 doubles every length and inverts the figure. Negative factors do not arise in models or maps, where only physical size is being described.
How do I scale a drawing up or down by a percentage?
Convert the percentage to a factor by dividing by 100. Enlarging to 150% is a factor of 1.5; reducing to 80% is a factor of 0.8. Note that reducing by 20% and then enlarging by 20% does not get you back to the original:
0.8 × 1.2 = 0.96, so you end up 4% smaller. To reverse a scaling exactly you need the reciprocal factor, not the same percentage the other way.