MoapaPk wrote:Some people define the grade of a slope as the 100%*tangent; others as the 100%*sine.
From DayHiker's mention of hypotenuse, I'm guessing that he is using the latter definition. Not too much difference when the angle is small, but a lot of difference when the angle goes to pi/2. Ze's question "...when the grade gets close to 100%, and you're scrambling up?" suggests that he is using the tan approach, as 100% in the sine definition is a vertical wall.
http://en.wikipedia.org/wiki/Grade_%28slope%29
No!!!!
Grade is defined as rise over run -- the tangent of the incline angle, NOT the sine.
And the tangent is what I am using! I mention hypotenuse because we were talking about trail distance, and that is the hypotenuse, not the horizontal distance. A hiker's feet and legs have no direct perception of the horizontal distance; a hiker travels the actual trail distance, and that, along with the elevation gain, is what is perceived by the hiker. For an analysis of the statistics of a hike, the actual horizontal distance traveled is quite meaningless in itself.
Check out the highlighted text:
Day Hiker wrote:I have only climbed San Antonio in winter via the Baldy Bowl, so I haven't been on the entire trail, but it's my understanding that the trail is only about 4 miles, not 8. Of course the elevation gain is the same either way, at around 3900 feet.
So, assuming a constant grade for simplicity, with a hypotenuse of 4 miles and a gain of 3900 feet, the trail's average grade is about 18.8%.
. . .
[With 3900 feet of gain,] to have only a 10% grade, the trail would have to be about 7.4 miles long, but it's not.
. . .
If you check those numbers, you will see that I am defining grade as the tangent of the incline angle, not the sine.
For the first case, if I were using the sine, the calculation of 3900 feet divided by 4 trail (hypotenuse) miles would yield around 18.5% which is not the number I posted.
For the second case, it's not so obvious because either definition would yield 7.4 miles, if rounded to the first decimal place. To four places, the correct value, defining grade as the tangent, is 7.4232 miles, which is how I did my calculation. Using the incorrect sine definition for the calculation would yield 7.3864 miles (39000 feet).