Slope, Grade, Degrees and Ratio: How to Calculate Each and Convert Between Them

Rise over run, and nothing else

Every slope figure starts from one fraction: the vertical rise divided by the horizontal run. Percent grade multiplies that fraction by 100, the angle takes its arctangent, and the ratio inverts it. Because all three come from the same fraction, you can always get from one to another without measuring again — but you cannot get from one to another by scaling, because the arctangent is not a straight line. Near zero the relationship is almost linear, at roughly 0.573 degrees per percent, which is why a 1 percent grade is 0.57 degrees and a 5 percent grade is 2.86 degrees, very nearly five times as much. By 50 percent that approximation would predict 28.6 degrees and the truth is 26.57.

The run in the fraction is the horizontal distance, not the length along the ground. Pull a tape up a bank and you measure the hypotenuse, which is always longer than the run beneath it. On gentle slopes nobody cares: at 10 percent the tape reads only 0.5 percent long. On a steep bank it decides the job. At 50 percent the sloped length is 11.8 percent longer than the run, and at 100 percent it is 41.4 percent longer. Anyone ordering turf, membrane or paving by a tape measurement taken along a 1-in-1 bank will be a third short. Measure the run with a level and a plumb line, or measure the slope length and divide by the square root of one plus the grade squared.

The ratio is written backwards depending on the trade

A ramp described as 1:12 and a bank described as 12:1 are the same slope. Accessibility codes write rise first — one unit up for twelve along — while earthworks and highway drawings write run first, twelve along for one up. Nothing in the notation tells you which convention is in use, and reading 12:1 as rise-over-run gives a grade of 1,200 percent, an angle of 85 degrees and a very confused excavator driver. The safe habit is to convert every ratio you are given into a percentage the moment you receive it, because a percentage cannot be read backwards.

Roof pitch adds a third notation. American roofers quote rise per twelve of run, so a 6-in-12 roof is 6 ÷ 12 = 50 percent and 26.57 degrees, and a 12-in-12 roof is 100 percent and 45 degrees. That is the ratio convention with the run fixed at twelve rather than at one, which makes pitches directly comparable but only within the same system. Convert the pitch to a percentage before you compare it to a drainage fall or a ramp gradient, or you will be comparing a fraction of twelve with a fraction of a hundred.

What the codes actually require

For ramps the numbers are legal, not advisory, and they come in pairs: a maximum gradient and a maximum rise before a landing. The ADA Standards cap a ramp at 1:12, that is 8.33 percent or 4.76 degrees, and allow at most 30 in of rise in one run — which at 1:12 is 30 ft of ramp — before a level landing at least 60 in long. Below 1:20, 5 percent, the surface is a walking surface rather than a ramp and those rules do not apply at all. In the UK, Approved Document M takes a different route to the same idea: 1:12 is permitted only over a going of 2 m, 1:15 up to 5 m and 1:20 up to 10 m, with no more than 500 mm of rise in a single flight.

Drainage falls run the other way — they are minimums, and they are small. The usual instruction to fall the ground away from a building is around 2 percent, which is 1.15 degrees and reads as almost nothing on a spirit level; the surveyor's version, a quarter inch per foot, is 2.08 percent. Flat roofs are typically specified at 1 percent or more so that ponding does not defeat the membrane. Because these figures are so shallow, they are the ones most often lost on site: a 2 percent fall across a 16 ft patio is only 4 in of drop end to end, and half of it disappears into the tolerance of a screed. Set them with a level and a graduated block, never by eye.

Frequently asked questions

Is a 100 percent slope vertical?

No. A 100 percent grade rises one unit for every unit forward, which is exactly 45 degrees — steep enough that you would scramble rather than walk, but only half the way to vertical in angle. Grades keep going: 200 percent is 63.43 degrees, 1,000 percent is 84.29 degrees, and true vertical is undefined because you divide by a run of zero. The intuition that 100 percent must be the maximum comes from percentages of a quantity, where 100 percent is all of it. A grade is not a percentage of anything; it is a ratio that happens to be written with a percent sign.

How do I measure a slope without a survey level?

A straight board, a spirit level and a tape will do it. Rest one end of the board on the high point, raise the other until the bubble centres, and measure straight down from the raised end to the ground: that drop is the rise and the board length is the run. A 4 ft board with a 4 in drop is 4 ÷ 48 = 8.3 percent. Repeat along the length rather than trusting one reading, because ground is rarely a plane and the average of three spans tells you more than one lucky span. A smartphone inclinometer will give you the angle directly, but check it against the board method once — laid on a bare floor it can read a degree out from the case alone.

Is percent grade the same as a percentage of the angle?

No, and the two only look alike near the bottom of the scale. Percent grade is the tangent of the angle multiplied by 100, and the tangent curve leaves the straight line quickly. Up to about 10 percent you can divide by 1.75 for a rough angle and be within a tenth of a degree: 10 percent gives 5.71 and the shortcut gives 5.71. By 25 percent the shortcut says 14.3 against a true 14.04, and by 50 percent it says 28.6 against a true 26.57. Above a tenth of a rise, use the arctangent properly or read the value from a table.