There's quite a fuss being made over on Trains.Com right now, on the matter of degrees and percent, and the like. Probably much more fuss than needs to be.
Here's how it works. In North America, grades are measured in percent, not angular degrees. Surveyors would measure segments of the right of way using the Surveyor's Chain, usually 100 feet long. When going up or down a grade, the difference in elevation between each end of the chain is recorded (again, usually in feet), and since this can be expressed against that 100 foot run, the grade is in per cent. A one foot rise over 100 feet is a one percent grade, and so on.
As an aside: yes, strictly speaking, grade is supposed to be computed using the horizontal edge of a triangle, not the "hypotenuse", which is what surveyors ended up doing. But for very slim triangles, the difference in length between these two sides of a triangle was insignificant: a 2 percent grade—steep in railroad terms—is just a bit over 1 degree from horizontal. At such angles, the difference in length between the hypotenuse and the horizontal edge is about .999, or less than one part per thousand.
|Rise (difference in elevations|
|Run (level distance between points)|
Surveyors were reluctant to use degrees for grade because of the extra math it involved. Nowadays, portable computers or even hand calculators make short work of all that trigonometry, but back then, it all had to be done using tables. Noting grade in terms of percent was comparatively simple. If you knew you had to clear a 1000 foot pass, and knew you had to keep the grade to no more than 1 percent, it would require a roadbed that was 1000 x 100 or 100,000 feet on either side of the summit. Since a surveyor's chain was 100 feet, it would require 1000 of those segments.
In modeling terms, measuring grade can be just as simple. All you need is a level, a piece of wood say 25 inches long, and another piece of wood with markings at fine regular intervals (say ⅛" or so). Since 25 is one quarter of 100, each quarter of an inch rise over 25 inches run is one percent; ⅛" is half of ¼", so it's half a percent, and ½" is twice ¼", so it's two percent.