Quick slope utility: enter rise and run to get angle, percent grade, ratio, 1-in-N, and the diagonal length.
The same slope can be written as an angle, a percent grade, a ratio, or as "1 in N". This utility converts between them and gives the diagonal length by the Pythagorean theorem. Useful for stairs, ramps, handrails and grading.
The four ways of writing a slope each suit a different trade. Angle (degrees from horizontal) is how stairs and roofs are usually discussed. Percent grade is the convention for roads, ramps, and site drainage — a 5% ramp rises 5 units for every 100 of run. "1 in N" is common in plumbing falls and accessibility ramps (the ADA limit is 1 in 12). The diagonal is the true travel length along the slope, which is what you measure when ordering a stringer, handrail, or rafter.
A quick sanity check: a 45-degree slope is exactly 100% grade and 1 in 1, because rise equals run. Anything steeper than 45 degrees passes 100%, which surprises people expecting percent to top out at 100.
The angle of a staircase is the arctangent of the rise over the run. For the slope of a single step (the true pitch a stringer is cut to) use the unit rise and unit tread; for the overall slope use total rise and total run.
A 7.5-inch riser with a 10-inch tread gives atan(7.5 / 10) = 36.9 degrees. Most comfortable domestic stairs land between 30 and 38 degrees; codes cap the steepest stairs around 42 to 50 degrees depending on the standard. To size and check a full flight by angle, use the straight stair calculator.