How to convert a hike's grade from percent to degrees
Grade as a percentage and slope as an angle in degrees are two different units for the same steepness, the way miles and kilometres are two different units for the same distance — except that converting between grade and degrees does not scale in a straight line the way a distance conversion does. A 20 percent grade is nowhere near a fifth of the way to vertical, and a 50 percent grade sounds twice as steep as a 25 percent one without the angle behind it being anywhere close to double. The conversion itself is one line of trigonometry. Getting a feel for why it behaves the way it does is the part worth sitting with.
The short answer
To turn a percentage grade into degrees, divide the grade by 100 and take the arctangent of the result. A 20 percent grade is arctan(0.20), which comes out to about 11.3 degrees. To go the other way, take the tangent of the angle and multiply by 100 — a 15-degree slope is tan(15°) × 100, or about 26.8 percent grade.
Neither direction needs anything more than the arctangent and tangent functions on an ordinary scientific calculator or a phone calculator app, switched to degree mode rather than radian mode.
Why the two numbers do not move together
Grade is rise over run — the same ratio a hike report already builds elevation gain and distance from, just written as a percentage instead of a fraction. An angle in degrees measures rotation, not a ratio, and the relationship between the two is a tangent function, not a straight-line scale. That is what makes them drift apart as a slope gets steeper: grade running from 0 to 100 percent goes from flat ground to a slope where rise equals run, while the angle covering that same range only reaches 45 degrees — a little more than half of the 90 degrees a genuinely vertical face would need.
The mismatch grows the steeper things get. Going from a 10 percent grade to a 20 percent grade roughly doubles both numbers, from about 5.7 to 11.3 degrees — close enough that the two feel interchangeable on gentle ground. Going from 100 percent to 200 percent, though, only moves the angle from 45 degrees to about 63.4 degrees, nowhere near double. Grade keeps climbing in a straight line as a slope steepens; the angle bends over and creeps toward 90 degrees without ever reaching it, since a slope a person can still walk up never quite becomes a sheer wall.
A worked example, from a hike report to degrees
Say a hike report shows 3.2 kilometres of distance and 480 metres of elevation gain for one steady climb. Grade first, the same division covered in how to calculate average grade: 480 ÷ 3,200 = 0.15, or 15 percent. That decimal is already the input the arctangent needs — no further division required. Arctan(0.15) comes out to about 8.5 degrees, so a climb a hiking app calls a 15 percent grade is, on a sign that talks in degrees instead, roughly an 8.5-degree slope.
Going the other way starts from an angle instead of a distance and elevation figure. A trailhead sign warning of a 12-degree ascent ahead converts by taking the tangent of 12 degrees, about 0.213, and multiplying by 100 for a 21.3 percent grade — steeper than the number "12" suggests to anyone expecting grade and degrees to line up one for one.
A quick reference table
- 5% grade ≈ 2.9°
- 10% grade ≈ 5.7°
- 15% grade ≈ 8.5°
- 20% grade ≈ 11.3°
- 30% grade ≈ 16.7°
- 50% grade ≈ 26.6°
- 100% grade = 45°
- 10° ≈ 17.6% grade
- 20° ≈ 36.4% grade
- 30° ≈ 57.7% grade
Why 100 percent does not mean vertical
The biggest source of confusion is treating a 100 percent grade as the halfway point to a sheer drop, or worse, as vertical itself. It is neither. A 100 percent grade means the ground climbs exactly as much as it runs — one metre up for every metre forward — which works out to a 45-degree slope: a steep scramble, and a long way from a cliff face. Grade only starts to describe something resembling "up" faster than "across" once it climbs well past 100 percent; the percentage figure is unbounded and can read in the hundreds for a genuine cliff, while degrees are capped at 90 no matter how steep the ground actually gets.
This is also why grade and degrees swap which one sounds more alarming depending on how steep the ground already is. On gentle terrain, the percentage figure looks larger and more dramatic than the degree figure for the identical slope — an easy grassy rise phrased as an "8 percent grade" sounds steeper than the same rise phrased as "4.6 degrees." On genuinely steep ground the relationship flips: the degree figure caps out near 90 while the percentage figure keeps climbing well past 100, so which unit sounds scarier depends entirely on where the slope in question sits on the curve.
Where TrailCam fits in
TrailCam reports distance and elevation gain for a route, with the elevation profile drawn underneath a replayed hike — the same two numbers the grade calculation above starts from. Converting a specific stretch to degrees is a manual step on top of that, useful when comparing a hike against a trail sign, a piste rating, or anything else phrased in degrees rather than percent, rather than a figure TrailCam calculates on its own. TrailCam is free to download.
Questions
- Is a 100 percent grade the same as a vertical, 90-degree slope?
- No. A 100 percent grade is a 45-degree slope — the ground rises exactly as much as it runs. A slope would need an unbounded grade percentage to approach vertical, since the tangent function connecting the two grows without limit as the angle nears 90 degrees.
- Why does a gentle slope look steeper described in percent than in degrees?
- Because grade and the angle it describes only stay close at low angles, and even there grade edges ahead — a 10 percent grade is about 5.7 degrees, so the percentage figure reads larger than the degree figure for the same gentle slope. The gap widens further as the slope gets steeper.
- Can I convert grade to degrees without a calculator?
- Not with real precision — arctangent is not something to work out by hand — but the reference table above covers the values that come up most often on a hike, and a phone calculator's arctan or tan function, set to degree mode, handles anything the table does not.
Recording your own routes
TrailCam is an iPhone app that records your route with GPS, keeps video clips, photos and voice notes pinned to the spot where you took them, and exports the result as GPX 1.1 or CSV. It is free to download.