Slope Percentage to Degrees: What Is a 10% Grade in Degrees? Formula and Conversion Chart
A road sign says 10%, a ramp is 1:12, a roof is 4 in 12, and a floor is 6/1000. Slope is written differently in every field, which makes comparing them confusing. This guide explains how the five ways of expressing slope relate to each other, the formula for converting percent grade to degrees, a chart of percent, 1:n ratio, per mille and degrees, typical slopes on roads, railways and car parks, and how to do the calculation in Excel or on a scientific calculator.
Five ways to express slope, and where each is used
However it is written, slope always describes the same ratio: how much the height changes for a given horizontal distance. Different fields have simply settled on the notation that suits them, so once you convert any of them to a single value (rise ÷ run), they can all be converted into one another.
| Notation | Meaning | Example | Typical use |
|---|---|---|---|
| Angle (degrees) | The angle from horizontal | 5.71° | Surveying, machinery, ski slopes |
| Percent (%) | Rise ÷ run × 100 | 10% | Road hills, car park ramps |
| Sun (Japanese roof pitch) | Rise per 10 units of run | 1 sun | Roofs in Japan (similar to rise per 12 in imperial roof pitch) |
| 1/n (ratio) | Rise 1 for every n of run | 1/10 (1:10) | Ramps, drain pipes, floor falls |
| Per mille (‰, mm/m) | Rise ÷ run × 1000 | 100‰ | Railways, floor levels, drainage |
Every example in the table above is the same slope. A 10% hill matches a 1-sun roof, a 1:10 ramp and a 100‰ railway line. As an angle it is about 5.71°.
The formula for converting percent and degrees
First convert the slope to rise ÷ run (call it the gradient value). 10% is 0.1, 1:12 is 0.0833…, a 4-sun roof is 0.4 and 6/1000 is 0.006. From there you can move to each notation with these formulas.
| What you want | From the gradient value | Example (gradient value 0.1) |
|---|---|---|
| Angle | arctan(gradient value) | 5.71° |
| % | gradient value × 100 | 10% |
| Sun | gradient value × 10 | 1 sun |
| n in 1/n | 1 ÷ gradient value | 1/10 |
| Per mille (mm/m) | gradient value × 1000 | 100‰ (100 mm per metre) |
To go from an angle back to a gradient, use tan: gradient value = tan(angle). For example, 5° gives tan(5°) ≈ 0.0875, which is about 8.75% or 1:11.4.
Why 100% is 45°, not 90°
A 100% slope sounds like a vertical wall, but 100% means the rise equals the run, which is 45°. Vertical (90°) rises without any horizontal distance at all, so in percent it would be infinite. Percent and degrees are not proportional: as a slope gets steeper, the percentage climbs much faster than the angle.
For gentle slopes up to about 10% (about 5.7°), though, you can treat them as roughly proportional. One degree is about 1.75%, so you can estimate in your head with "degrees × 1.75 ≈ %" and "% ÷ 1.75 ≈ degrees". 3° is about 5.2% (exactly 5.24%) and 8% is about 4.6° (exactly 4.57°). From around 20% the error becomes noticeable, so use the formula or the chart when you need an accurate figure.
Conversion chart: percent to degrees, 1/n and per mille
| % | Degrees | 1/n | Rise per metre | Everyday example (Japanese standards) |
|---|---|---|---|---|
| 0.5% | 0.29° | 1/200 | 5 mm | Minimum fall for large drain pipes |
| 1% | 0.57° | 1/100 | 10 mm | Drain pipes (75 and 100 mm), floor falls on concrete |
| 2% | 1.15° | 1/50 | 20 mm | Small drain pipes, balcony floor falls |
| 5% | 2.86° | 1/20 | 50 mm | A gentle, easy-to-walk ramp |
| 6.67% | 3.81° | 1/15 | 66.7 mm | Recommended standard for outdoor ramps |
| 8.33% | 4.76° | 1/12 | 83.3 mm | Mandatory maximum for accessible ramps |
| 10% | 5.71° | 1/10 | 100 mm | A hill often seen on steep-gradient warning signs |
| 12.5% | 7.13° | 1/8 | 125 mm | Upper limit for ramps that replace stairs |
| 17% | 9.65° | 1/5.9 | 170 mm | Upper limit for car park driveways |
| 20% | 11.31° | 1/5 | 200 mm | A 2-sun roof |
| 30% | 16.70° | 1/3.3 | 300 mm | A 3-sun roof |
| 50% | 26.57° | 1/2 | 500 mm | A 5-sun roof |
| 100% | 45.00° | 1/1 | 1000 mm | A 45-degree roof (10 sun) |
Very small slopes, such as a floor that is out of level, are easiest to read in per mille. 3/1000 is about 0.17°, 6/1000 is about 0.34° and 10/1000 (1/100) is about 0.57°.
Typical slopes on roads, railways and car parks
In Japan, the maximum longitudinal gradient of a road (how steeply it climbs or descends) is set by the Road Structure Ordinance according to the design speed. Faster roads must be gentler, and where the terrain leaves no choice, steeper exceptional values are allowed.
| Design speed | Maximum gradient | Exceptional value | Angle (maximum) |
|---|---|---|---|
| 120 km/h | 2% | 5% | About 1.15° |
| 100 km/h | 3% | 6% | About 1.72° |
| 80 km/h | 4% | 7% | About 2.29° |
| 60 km/h | 5% | 8% | About 2.86° |
| 40 km/h | 7% | 10% | About 4.00° |
| 20 km/h | 9% | 12% | About 5.14° |
- Road signs: steep ascent and steep descent warning signs show the gradient as a percentage, such as 10%. Cycling up a 10% hill feels very hard.
- Car parks: in Japan, the Parking Lot Act Enforcement Order limits the driveway slope of larger off-street car parks to 17% (about 9.65°) or less.
- Railways: railway gradients are given in per mille (‰), and most lines run at a few ‰ up to about 30‰. The 80‰ (about 4.6°) on the Hakone Tozan Railway is known as the steepest gradient in Japan for a conventional railway that runs on rail-wheel adhesion alone.
How to calculate it in Excel, on a scientific calculator or on your phone
| What you want to do | Excel or Google Sheets formula | Scientific calculator |
|---|---|---|
| % → degrees (% in A1) | =DEGREES(ATAN(A1/100)) | tan⁻¹(% ÷ 100) in DEG mode |
| Degrees → % (angle in A1) | =TAN(RADIANS(A1))*100 | tan(angle) × 100 |
| 1/n → degrees (n in A1) | =DEGREES(ATAN(1/A1)) | tan⁻¹(1 ÷ n) |
| Rise and run → degrees | =DEGREES(ATAN2(run,rise)) | tan⁻¹(rise ÷ run) |
Excel’s ATAN and TAN work in radians, so remember to convert with DEGREES and RADIANS. On a scientific calculator, check that the angle unit is set to DEG (degrees) before you start. If it is left in RAD, you will get a completely different answer.
Do every slope conversion in one go
In the Convert slope tab of the slope and angle meter (a phone spirit level), enter any one of degrees, percent, sun, 1/n or per mille and it shows all the other notations together with the slope length factor, and compares the result with floor, drainage, ramp and car park standards. The Slope chart tab gives a lookup table by sun, percent, 1/n and degrees, and if you open it on your phone you can measure the hill or surface in front of you and read it straight off in percent or degrees.
Roof pitch in sun is covered in more detail in what angle is a sun roof pitch, the length a 1:12 ramp needs in how long a 1:12 ramp has to be, and the 1/100 fall for drain pipes in how many millimetres a 1/100 drain fall is.
FAQ
- How many degrees is a 10% slope?
- About 5.71°. Work it out with angle = arctan(10 ÷ 100). It is the same slope as 1:10, a 1-sun roof and 100‰ (100 mm per metre).
- Is a 100% slope vertical?
- No, it is 45°. 100% means you rise as far as you travel horizontally. A vertical 90° slope would be infinite in percent.
- How do I convert degrees to percent?
- Use % = tan(angle) × 100. 1° is about 1.75%, 5° is about 8.75% and 30° is about 57.7%. For gentle slopes you can estimate it in your head with degrees × 1.75.
- What percentage is a 1:12 slope?
- About 8.33%, or about 4.76°. Work it out with 1 ÷ 12 × 100. 1:15 is about 6.67%, 1:20 is 5% and 1:100 is 1%.
- What does ‰ (per mille) mean?
- It means parts per thousand: rise ÷ run × 1000. It is the same number as how many millimetres the slope rises per metre, so 10‰ is 10 mm per metre, which is 1% or 1/100. It is often used for railway gradients and floor levels.
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