Slope: percent, degrees, ratio
slope (%) = 100 × rise / horizontal distance
The same slope in its four writings: the percentage of road signs, the angle in degrees, the 1:n ratio of accessibility ramps, and the rise per hundred metres travelled. The input is whichever you have: the percentage, the angle, or the rise-distance pair measured on the ground. The tool dismantles the most widespread misconception of the topic, 100% is not vertical but 45°, and states what the percentage really compares: the rise to the horizontal distance, not to the road.
10 % · 5.7106°
- Calculation
- angle = atan(10 / 100) = 5.7106°
- Ratio
- 1:10
This slope is 10 %, that is an angle of 5.7106°.
The classic misconception: 100% is not vertical, it is 45°, the angle where you climb as much as you advance. The percentage is a tangent, it grows without bound: 200% makes 63°, vertical would be an infinite slope.
The slope compares the rise to the HORIZONTAL distance, not to the road: on the ground you measure the road (the hypotenuse), slightly longer. The difference stays under 1% up to a 14% slope: for usual roads and trails, confusing the two is harmless, for a roof it is not.
Landmarks: 5% is the accessibility-ramp standard (1:20), 10% a steep-hill road sign, 20% the steepest streets in Europe, 100% the diagonal of the square. A "45°" roof is at 100%; roof pitches are in fact often given in percent.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasslope (%) = 100 × rise / horizontal distance
slope (%) = 100 × rise / horizontal distance
angle = atan(slope / 100)
ratio 1:n with n = 100 / slope
road = √(distance² + rise²)
The slope in percent is one hundred times the tangent of the angle: the metres climbed per hundred metres advanced flat. The three writings convert exactly, and the road length comes from Pythagoras.
- Slope in percent
- · the rise per 100 m of horizontal advance. It is a tangent: it exceeds 100% as soon as the angle exceeds 45°.
- Ratio 1:n
- · one metre climbed per n metres advanced: the writing of ramps and accessibility standards. 5% = 1:20.
- Rise
- · the altitude difference between two points, the reading of maps and hiking GPS units.
- Horizontal distance
- · the flat distance, the map’s. The real road is the hypotenuse, always slightly longer.
Validity domainThe calculation is exact trigonometry: no approximation.
The calculation is exact trigonometry: no approximation. The only subtlety is the input convention: if your "distance" is measured along the road (wheel, GPS), the true slope is slightly higher than the naive quotient, the gap staying under 1% up to a 14% slope. Slopes count positive here, from 0° included to 90° excluded: the direction of travel (up, down) is trip information, not geometry.
The 100% misconceptionA 100% slope sounds like a wall: it is actually 45°, the diagonal of the square, the angle where every metre of advance costs a metre of climb.
A 100% slope sounds like a wall: it is actually 45°, the diagonal of the square, the angle where every metre of advance costs a metre of climb. The mistake comes from reading the percentage as a fraction of vertical, when it is a tangent: the ratio of two lengths, growing without bound. 200% makes 63°, 1000% makes 84°, and vertical has no percentage, its tangent not existing. Hiking GPS units that briefly show "60%" on a steep pitch are therefore not announcing climbing: it is 31°.
Reading the resultThe 1:n ratio is the writing builders speak: a 1:20 ramp climbs one metre in twenty, and it is the usual accessibility threshold for a path without landings.
The 1:n ratio is the writing builders speak: a 1:20 ramp climbs one metre in twenty, and it is the usual accessibility threshold for a path without landings. The road length per hundred flat metres shows the gap between map and ground: negligible on a road (100.5 m at 10%), substantial on a roof (141 m at 100%). The note’s landmarks place your result between the gentle ramp and the steepest street.