One item of frequent discussion in ski & snowboard circles is the steepness of a given slope. However, I get the feeling when I see and hear discussions on slope of a given hill is people are either not using the terminology in the same way as others or they use the terminology in a sloppy fashion.
First, let us consider the below triangle:
This is 90°-45°-45° angle:
α = β = 45°
γ = 90°
A=1000 units (length)
B=1000 units (length)
When I speak of slope of a ski hill I think of the angle α in the above diagram.
Another common term used to describe the slope of a snowboard run is in terms of percent grade. The percent grade of a slope is defined as: 100 x (the change in vertical elevation divided by the change in horizontal distance). So, in the above diagram we drop 1000 units of vertical in 1000 units of horizontal = 100*(1000/1000) which makes for 100% grade. No, a grade of 100% is not straight down.
In fact, this is tangent of angle α so in order to find the angle α when given a percent grade (g) simply use your calculator to compute tan -1(g).
You will find, in general, most ski slopes we ski on are well under 100% grade.