Percent grade vs degrees: why they are not the same
How road signs, ramps, and surveys describe steepness, why a 100% grade is only 45 degrees, and how to convert between grade, angle, and ratio.
Four ways to say the same slope
A slope is just vertical rise divided by horizontal run, but that single relationship gets dressed up in four outfits. Percent grade multiplies rise over run by 100, so a road that climbs 8 meters over 100 meters is an 8% grade. The angle is the arctangent of rise over run, measured in degrees from the horizontal. The 1:N ratio flips the fraction and normalizes the rise to 1, so N is the run per unit of rise. Slope distance is the fourth value, the actual length you would walk along the incline. This tool takes any one of the first three and fills in the rest.
Why 100% is not straight up
A common surprise is that a 100% grade does not mean vertical. Percent grade is a ratio, not a fraction of a right angle, so 100% simply means the rise equals the run: one meter up for every meter forward. That is a 45 degree slope, and the tool confirms it with an angle of 45 and a 1:1 ratio. Because grade grows without bound while the angle can never reach 90, the two scales pull apart fast. A 50% grade is about 26.57 degrees, a 100% grade is 45 degrees, and a 200% grade is about 63.43 degrees.
Where each form is used
Highway and railway engineers favor percent grade because it adds intuitively over distance and appears on those triangular warning signs before a steep descent. Roofers and carpenters often use the ratio, quoting a roof as 4:12, meaning 4 units of rise for every 12 of run. Surveyors, machinists, and anyone with a protractor or digital level think in degrees. Wheelchair ramps are usually specified as a ratio too, with guidance commonly citing a maximum of 1:12, which is an 8.33% grade or about 4.76 degrees.
Converting by hand
To go from grade to angle, divide the grade by 100 and take the arctangent: atan(0.10) is 5.710593137 degrees for a 10% grade. To go the other way, take the tangent of the angle and multiply by 100. For the ratio, divide 100 by the percent grade, so a 10% grade is 1:10 and a 25% grade is 1:4. Slope distance needs an actual rise and run, then it is the Pythagorean hypotenuse, the square root of rise squared plus run squared. Rise 10 and run 100 gives a slope distance of 100.498756211.