A 100% grade is 45 degrees. Vertical is not 200% or 1000% — it has no percentage at all.
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Grade20%
Angle11.3099°
Roof pitch2.4:12
As a ratio1 in 5
Steep. Low gear in a car, and hard work on foot.
Measured against the run, or along the slope
Rise ÷ horizontal run
20%
Rise ÷ sloping distance
19.6116%
1.9804% apart at this angle. Road and roof conventions divide by the horizontal run; some older railway and survey practice divides by the distance actually travelled along the slope. Below about 5° the two agree closely enough to ignore; here they do not.
Work out the other sides
Rise2
Run10
Along the slope10.198
Same units in, same units out. The sloping distance is always longer than the run — which is why a roof needs more material than its footprint, and why a hill is further to walk than the map suggests.
For scale
8.33% · 4.8°Accessible ramp (ADA / BS 8300 maximum)Written as 1:12. Steeper than it sounds to push a wheelchair up, which is why the standards also cap the length of a single run.
2.5% · 1.4°Mainline railway (steep)Written as 1 in 40. Steel on steel has very little grip, so railway gradients are tiny by road standards and a 1 in 40 climb is considered severe.
6% · 3.4°Motorway maximum (typical design limit)Around 3.4°. Design limits vary by country and by design speed.
10% · 5.7°Steep road warning sign (UK / EU)The threshold at which a gradient sign usually appears — 5.7°.
35% · 19.3°Steepest public roads in the worldAbout 19°. Substantially steeper than anything on a normal road network.
33.3% · 18.4°Roof: 4:12 (shallow)18.4°. Around the minimum for most asphalt shingles.
50% · 26.6°Roof: 6:12 (conventional)26.6°. The most common domestic pitch in much of the world.
100% · 45.0°Roof: 12:12Exactly 45° — and exactly 100% grade, which is the clearest demonstration that 100% is not vertical.
40% · 21.8°Black ski run (typical)Around 22°. Ski runs are usually quoted in percent, which is why they sound steeper than the degrees suggest.
70% · 35.0°Staircase (typical domestic)Around 35°. Steeper than any road or roof most people encounter.
What this converter covers
Percent grade, degrees, roof pitch and “1 in N” from any one of them — plus rise, run and the sloping distance, which is the one you actually have to build or walk.
Percent, degrees, x:12 roof pitch and 1-in-N, all four at once
Entry from any of them, or from a rise and run directly
The third side of the triangle from any known length
Both percent conventions — against the run and along the slope
Reference grades from railway gradients to domestic staircases
Four conventions 100% ≠ vertical Roof pitch Reference grades
Planning support from the values you enter — not professional advice.
A 100% grade is 45 degrees
This is the single most common mistake about slopes, and it is an intuitive one: percentages usually run from 0 to 100, so a 100% grade sounds like the maximum — vertical, or something close to it.
It is 45 degrees. Percent grade is rise divided by horizontal run, so 100% means you rise one metre for every metre forward, which is a slope at exactly half a right angle.
And the scale does not stop there. It has no top at all. As the slope steepens, the run shrinks towards zero and the percentage climbs without bound:
Percent grade against the actual angle, as a slope approaches vertical
Grade
Angle
10%
5.7°
20%
11.3°
50%
26.6°
100%
45.0°
200%
63.4°
1000%
84.3°
Vertical
90° — no percentage exists
So a vertical face is not 100%, or 500%, or any number. The run is zero and you cannot divide by it. The calculator above refuses 90 degrees rather than inventing a figure, because inventing one would be the exact error the page is here to correct.
The practical version of this: a 20% road sign — steep enough that most countries put up a warning — is 11.3 degrees. Steep to drive, and about an eighth of the way to vertical rather than a fifth.
Three trades, three denominators
The other reason slopes are confusing is that four different notations are in everyday use, each dividing the rise by a different thing, and none of them signals that the others exist.
The same slope written four ways, and where each notation is used
Notation
Means
Used by
50%
Rise per 100 of run
Roads, ski runs, cycling
6:12
Rise per 12 of run
Roofing
1 in 2
Rise per N of run
Railways, drainage, accessibility
26.57°
The actual angle
Surveying, engineering, machining
Every row above is the same slope. A roofer’s “6:12” and a road engineer’s “50%” describe an identical pitch, and nothing about the two numbers suggests it.
The ratio forms invert in a way that catches people out, too. A bigger N in “1 in N” means a shallower slope: 1 in 40 is gentle and 1 in 4 is severe. So a railway gradient described as “1 in 30” is steeper than one described as “1 in 100”, which is the reverse of how percentages read.
The reason the trades differ is practical rather than perverse. Roofers work in feet and inches and a twelve-unit run makes a whole-number rise for common pitches. Railways deal in gradients so slight that percentages would be awkward fractions, while “1 in 200” is memorable. And roads settled on percent because a sign has to be read at speed.
Run or slope length — which distance?
There is a second ambiguity hiding under all four notations: the rise is divided by a distance, and there are two distances available.
The horizontal run is the distance as a map sees it — the base of the triangle. The slope length is the distance you actually travel, along the hypotenuse. Road, roof and modern survey practice divide by the run; some older railway and marine conventions divide by the distance travelled.
For gentle slopes it makes no practical difference. At 5 degrees the two figures differ by 0.38%, which is far inside any measurement error. But the gap grows quickly:
The two conventions compared as the slope steepens
Angle
Rise ÷ run
Rise ÷ slope length
Difference
1°
1.75%
1.75%
0.02%
5°
8.75%
8.72%
0.38%
10°
17.63%
17.36%
1.54%
30°
57.74%
50.00%
15.47%
45°
100.00%
70.71%
41.42%
That 41.42% at 45 degrees is exactly √2 − 1, which is a satisfying way to remember that the two conventions diverge by the diagonal of a square. Below about 10 degrees — which covers every road, ramp and railway gradient anyone is likely to meet — the distinction is safely ignorable, and that is why almost nobody knows it exists.
What a grade feels like
Percentages are hard to picture, so some anchors are worth carrying.
Below 2% is where railways live. Steel wheels on steel rail have very little grip, so a mainline gradient of 1 in 40 — 2.5% — counts as severe and needs planning around. A slope you would not notice walking is a serious engineering problem for a freight train.
Around 8% is the maximum for an accessible ramp, written as 1:12. It sounds gentle and is genuinely hard work to push a wheelchair up, which is why the standards also limit how long a single run may be before a landing.
10% is roughly where a road gets a warning sign. It is 5.7 degrees — barely tilted, on paper — and quite steep to cycle up.
30 to 50% covers difficult ski runs and shallow roofs. This is where the two units diverge most confusingly in conversation, because skiers quote percentages that sound alarming and roofers quote pitches that sound tame for the same angle.
Around 70% — 35 degrees — is a typical domestic staircase, which is a useful calibration: steeper than any road you have driven and steeper than most roofs, and you climb one every day without thinking about it.
Rise, run and the material you need
The slope itself is often not the number you want. What you actually need is a length — of rafter, of ramp, of path — and that is the sloping side, which is always longer than the horizontal run.
A 6:12 roof over a 10 metre run has a 5 metre rise and a rafter length of 11.18 metres. So the roof surface is 11.8% larger than the footprint it covers, and a materials estimate based on the floor plan is short by that much before any allowance for waste or overhang.
That factor grows with pitch, and quickly. At a 4:12 pitch the surface is 5.4% more than the footprint; at 6:12 it is 11.8%; at 12:12 it is 41.4% — the same √2 that turned up earlier, because a 12:12 roof is at 45 degrees. Which is worth knowing before ordering.
The same arithmetic works in reverse for a ramp. An accessible ramp climbing 750 mm at the 1:12 maximum needs 9 metres of horizontal run, and the ramp surface itself is 9.03 metres — for a ramp the two are nearly identical, because the slope is shallow, which is the mirror image of the roof case above. The calculator will complete the triangle from whichever side you happen to know.
Related calculators
Other geometry with more than one convention:
AngleDegrees, radians, gradians, arcminutes and arcseconds — and why a spreadsheet’s SIN(90) returns 0.894 rather than 1.
LengthMillimetres to miles on the exact 1959 factors, with the mil kept clearly apart from the millimetre — they differ 25-fold.
AreaSquare feet, square metres, acres and hectares, with the factors squared for you — a square metre is 10.76 sq ft, not 3.28.
Tyre SizeSidewall, diameter and rolling circumference from any tyre code — plus exactly how far a size change throws off the speedometer.
Aspect RatioExact ratio from any resolution, the missing dimension, and what letterboxing costs — flagging the resolutions that are not the ratio they claim.
Tap Drill and Drill SizeTap drills at any thread engagement plus the nearest bit in all four drill series — and why “subtract the pitch” leaves 92% of the thread.
The trigonometry needs no authority — a grade is a tangent and an angle is its arc. What the sources establish is which convention each trade actually uses, since that is the whole difficulty: the highway authorities sign percentages, the roofers write rise per twelve, the railways write one in N, and the accessibility standards write a ratio. Four notations, one quantity, and no indication on any of them that the others exist.
Manual on Uniform Traffic Control Devices — grade warning signsFederal Highway Administration, US · verified 2026-09-07 · That highway grades are signed as a percentage of rise over horizontal run, which is the convention this page treats as the road standard
This converts between notations for a slope; it is not a design tool. Anything that has to comply — an accessible ramp, a drainage fall, a roof that has to shed water, a road or a path — is governed by a standard or building regulation that specifies not only the maximum slope but usually also the maximum length of a run, the landings required, the cross-fall, the tolerance and the minimum as well as the maximum. A gradient that satisfies the number here can still fail the specification. Two further limits worth naming. The reference grades in the table are typical or indicative figures for orientation, not design values, and they vary by country, by design speed and by the specific standard in force. And this page treats a slope as a single constant gradient; real ground, real roofs and real roads change slope along their length, so a measured average tells you little about the steepest point, which is usually the one that matters.
Published the Slope and Grade Converter: percent grade, degrees, roof pitch and 1-in-N ratios, entered from any of them or from a rise and run directly.
States plainly that a 100 percent grade is exactly 45 degrees and that vertical has no percentage at all -- the horizontal run goes to zero, so the value is undefined rather than large. The engine refuses 90 degrees rather than inventing a figure, which is the error the page exists to correct.
Sets the four trade notations side by side, since a 6:12 roof and a 50 percent road are the same slope written unrecognisably differently, and the 1-in-N form inverts so that a bigger N is a shallower gradient.
Returns both percent conventions -- rise over horizontal run and rise over the distance actually travelled -- because they agree to within 0.38 percent at 5 degrees and differ by 41.42 percent at 45, which is exactly root 2 minus 1.
Completes the triangle from any known side, which is usually the number actually wanted: a 6:12 roof over a 10 metre run needs 11.18 metres of rafter, so the surface is 11.8 percent larger than the footprint.
Includes reference gradients from mainline railway limits to domestic staircases, because a bare percentage is not something most people can picture.
Verified by 76 automated cases, including that all five entry points agree on the same slope and that every completed triangle satisfies Pythagoras.
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