Slope is rise over run, and the same slope gets written three ways: as a decimal or fraction, as a percentage grade, and as an angle in degrees. A 1-in-12 ramp is a slope of 0.0833, a grade of 8.33 per cent and an angle of 4.76 degrees. Grade is the tangent of the angle expressed as a percentage, so a 100 per cent grade is 45 degrees rather than vertical — which is the single most common misreading of a road sign.
The three notations belong to different trades. Roads and railways use grade, roofing uses a ratio, and anything drawn or machined uses degrees.
How do the three compare?
The relationship is a tangent, so the notations diverge as the slope steepens. Below about 10 per cent, grade and angle track each other closely enough that people confuse them without consequence; above it they part company fast.
| Ratio | Grade | Angle |
|---|---|---|
| 1 in 20 | 5% | 2.86° |
| 1 in 12 | 8.33% | 4.76° |
| 1 in 10 | 10% | 5.71° |
| 1 in 4 | 25% | 14.04° |
| 1 in 2 | 50% | 26.57° |
| 1 in 1 | 100% | 45° |
| 2 in 1 | 200% | 63.43° |
A vertical drop has no grade at all. The run is zero, the division is undefined, and any tool that reports a number there is wrong rather than precise.
What does a road sign mean?
Grade, almost always, and it is a percentage of horizontal distance rather than of the distance travelled. A 10 per cent hill climbs one metre for every ten metres measured along the ground, which is an angle of 5.71 degrees and feels considerably steeper than that sounds.
The steepest public roads in the world sit around 35 per cent, which is 19.3 degrees. Anyone quoting a road as being "at 45 degrees" is describing a 100 per cent grade, which does not exist on a public highway and is roughly three times steeper than the steepest one that does.
Railways use much smaller numbers, often quoted in per mille — parts per thousand — because steel wheels on steel rail have little grip. A 1 in 100 gradient is 1 per cent, 10 per mille and 0.57 degrees, and that is already a significant climb for a heavy train.
What about roof pitch?
Roofing in imperial markets uses rise over a fixed run of 12, written as 6:12 or "six in twelve". That is a slope of 0.5, a grade of 50 per cent and an angle of 26.57 degrees.
The fixed denominator is what makes the notation useful on site: comparing 4:12 with 9:12 is comparing two numerators, and no arithmetic is needed. It also means a roof pitch cannot be read as a ratio in the civil engineering sense, where 1 in 12 means the opposite arrangement of the same two numbers.
Metric roofing generally uses degrees, which is unambiguous and does not survive translation into the imperial convention without conversion.
Why do accessibility ramps use 1 in 12?
Because it is shallow enough to self-propel and short enough to build. A 1-in-12 ratio — 8.33 per cent, 4.76 degrees — is the maximum in several national accessibility standards, and it means a 750 mm rise needs 9 metres of ramp before any landings are added.
That length is why the ratio matters so much in practice. Halving the gradient to 1 in 24 doubles the run to 18 metres, and the space required is usually the binding constraint rather than the geometry.
Ski runs use a fourth convention again — colour grading rather than a number — and the underlying gradients vary by country, so a red run in France and a red run in Austria are not a promise about the same angle. Where a figure is quoted it is usually a maximum grade rather than an average one, which is why a run rated at 40 per cent can be comfortable for most of its length.
Does the order of the points matter?
Not to the slope. Working from (1, 2) to (4, 8) gives a rise of 6 over a run of 3, and reversing the points gives −6 over −3 — the same slope of 2, because both signs flip together.
What the sign does tell you is direction: a positive slope rises left to right, a negative one falls, and a slope of zero is level. On a road that distinction is the difference between a climb and a descent, and the sign on the sign is doing real work.
Questions people ask
What if the two x values are the same? The line is vertical, the run is zero, and the slope is undefined. It is not infinite and it is not zero — division by zero has no value.
How is the distance worked out? By Pythagoras on the rise and the run. A 1-in-12 ramp with a 12-unit run has a surface length of 12.04 units, which is why the ramp is always slightly longer than the floor space it covers.
Is grade the same as gradient? In common use, yes. In mathematics, gradient can mean the slope of a line or a vector of partial derivatives, so the word carries more meanings than the road sign does.
What is per mille? Parts per thousand, written ‰. A gradient of 25‰ is 2.5 per cent, and it is the standard notation on railways across much of Europe.
Grade for roads, ratio for roofs and ramps, degrees for anything drawn — and a tangent between all three. The slope calculator reports all of them at once from two points, the volume calculator handles the geometry that usually follows, and the scientific calculator does the trigonometry if you would rather see the working.