The following exercises were adapted from the sources listed the References section.
A tangent is the ratio of the height (rise) divided by the base (run) of a right triangle. A grade is a tangent multiplied by 100, also known as percent slope.
Slope, tilt, or inclination can be expressed as a rise/run ratio, grade (percent slope), or an angle. Of these three ways, the angle is the less informative one as it does not give you much of an idea how the rise compares to the run.
By contrast, stating a ratio such as 1 in 20 tells you that for every 20 horizontal units traveled, the altitude increases 1 unit. Stating this as a grade, whatever horizontal distance you travel, your altitude increases by 5%. Using this information:
If a roof slope is 38 degrees: what is the rise per foot of run?
Suppose you are building a house. The building width is 30 feet with a roof slope as in the previous exercise. How much space do you have overhead in the center of the attic?
If you are 6-feet tall and are standing at the center of the mentioned attic: how far you can walk towards the eaves before whacking your head?
For pedestrian facilities on public access routes, the maximum accepted running grade of sidewalks is 5%. Suppose you build a 100-feet long sidewalk up the slope of a hill and the low end to the high end of the walk is four feet. Are you in compliance with the building department inspection requirements? Why?