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# How do you Calculate Degrees of a Slope?

Okay, so I'm on google earth and want to figure out the degrees of a trail or section of a trail. How do I do this?

Such as on Annapurna at Hunter, I found a 20 foot vertical section and if I were to put it into a triangle, then the hypotunese would be 41. I'm stuck now!!! Help please, and please please please gimme the formula!

My guess is to use ?trigonometry?? And then I'm stuck. Once again...

HELP!!! lol
Quote:
 Originally Posted by skskskier Such as on Annapurna at Hunter, I found a 20 foot vertical section and if I were to put it into a triangle, then the hypotunese would be 41. I'm stuck now!!! Help please, and please please please gimme the formula!

sin( theta ) = opposite/hypotunese, where theta is the angle above the horizontal and the hypotunese

theta = arcsin( opposite/hypotunese ) = arcsin ( 20/41 ) ...
I'm going to take a shot at being a math whiz today and use the standard Windows Calculator in Scientific view.

First calculate the slope in percent 20/41=.4878 or 48.78%

Now click the INV (inverse) button and sin (sine) and the result is 29.2 degrees.

Edit: you are using google (or a map) to estimate the horizontal distance traveled, and are not actually measuring the hypotenuse which is the actual distance traveled on the ground. Therefore, the formula above will yield the correct result. If you know the vertical distance and use a tape measure along the ground, then the previous answers are correct. The actual distance traveled is 45.62 feet =SQRT(20^2 +41^2)
This is all sooooo confusing, maybe it might be because I don't have an inverse button, but can like show me how to solve the Annapurna one w/o the inverse button? Yes, I'm a confused little boy lol.
What's with the fascination with steepness in numbers with you.

I ski on mountains not angles or percent grade, I couldn't care less how steep it is. I look at it and if I think I can do it I give it a go.

If its at a resort, on a marked trail map and there are plenty of other people going down it. It isn't that special.
It's just fun to know lol.
I guess it is interesting to know for comparison purposes but when its steep, like really steep, I'm pretty sure I'll know it.
Okay yay after something like an hour straight on the internet I figured out what to do.

Height / Invisible line (Hypotunese) and then press this second button on my calculator and press cosine. I got 29.2 for Annapurna and a bunch of other stats I already knew were true so yay! O and don't think you guys didn't help... you sprung a lotta ideas into my mind which helped understand it better! Thank You So Much!
It's good to know the pitch of a slope in the Backcountry. the prime pitch for avalanches is about 37 degrees. There are very few under 30 degrees. A couple of degrees of difference doesn't seem like much to us, but it can be huge in avy terrain. I use an inclinometer in the field. You can also get a feel for pitch on topo maps using a scale. There is one printed on the Life Link inclinometer tool.
If you have the vertical drop and the true horizontal distance (from a map), then divide the vertical drop by the horizontal distance. This number is the tangent of the angle. Whatever angle gives you this number when you use the tangent button is the angle of the slope. Your calculator should allow you to press inverse then tangent to work back words instead of guessing angles and taking their tangents until you hit the mark.

If you have the vertical drop and the measured sloping distance then divide the vertical drop by this sloping distance (hypotenuse). This number is the sine of the angle. Whatever angle gives you this number when you use the sine button is the angle of the slope. Your calculator should allow you to press inverse then sine to work back words instead of guessing angles and taking their sines until you hit the mark.

Something worth remembering, the sine of 30 degrees is 1/2.

This is an old thread, but apparently it pops up in Google search results, causing people to repost this misleading and blatantly incorrect information elsewhere. Only Ghost posted the correct answer (and skierfanatic provided a proper link), while the rest of the thread is incorrect.

The skiing slope grade is calculated using the same standard approach used everywhere. The slope grade is the ratio between the altitude gain and the planar projection of the distance traveled. It is the ratio between the vertical cathetus and horizontal cathetus of the right triangle (not hypotenuse, as it is incorrectly stated in this thread), expressed as percentage of [0, 1] segment. It is the tangent of the slope angle, not sine of the slope angle.

In other words, in order to convert the grade percentage to degrees, one has to use 'arctan' (not 'arcsin') function of scientific calculator. 100% slope grade corresponds to 45 degree slope (just remember this relationship as a basic correctness test and it will help you to avoid embarrassing errors in the future).

I understand that 'arcsin' produces significantly more "impressive" degree values, but it has no connection to reality. Internet trolls often use this widespred misconception for their malicious purposes, which is why I was quite surprised to see posters ranked as high as "moderator" to promote that nonsense here.

Keep in mind that the pitch of a run taken off a topographic map and the pitch of that same run when it is filled in with 20 feet of snow can be very different. Plus, what matters more than the average pitch is the length and steepness of the steepest section. I have skied Main Chute at Squaw--53 degrees, according to Squallywood, accurately measured. Does this mean I can ski a 50 degree chute--no way. Main Chute is very short and very concave, the pitch eases with each turn, plus when I skied it there was a little track that traversed the steepest part of the entry. Or take an even more exteme case--many steep non expert runs have moguls with faces over 50 degrees. Hardly the same as skiing a sustained 50 degree face. (Another interesting feature of Squallywood is that a number of runs, because they are so short, are rated easier if you straightline, than if you make turns, including Main. But don't even think about straightlining a 50 degree 300 meter couloir in the Alps.)

A good website I learned about in Avy class is caltopo.com you can pull up usgs topo maps and then use a gradient overlay to easily identify slope angles.(there is also some mental math trick for ballparking slope angle based on 40' contour lines but I can't remember it) Keep in mind that the snowpack can effect the actual slope angle though... Your best bet is just to measue with an inclinometer.

SOH-CAH-TOA dude...

Quote:
Originally Posted by bcohen5055

A good website I learned about in Avy class is caltopo.com you can pull up usgs topo maps and then use a gradient overlay to easily identify slope angles.(there is also some mental math trick for ballparking slope angle based on 40' contour lines but I can't remember it) Keep in mind that the snowpack can effect the actual slope angle though... Your best bet is just to measue with an inclinometer.

Great link to a really interesting tool.  I'm not sure I follow what you mean by "use a gradient overlay though"  The "slope angle shading" is interesting but not precise.  The "DEM shading" "layer"  offers more precision but might take some time for me to figure out how to use it best...  Any suggestions?

It looks like the tool is based solely on USGS data which has 40' contours in most mountain areas.  Accuracy will be limited to slopes with at least 40' of drop.  Despite that limit, this tool has more "out of the box" than anything else I've seen.

Any bugs with this tool?  (I have found a lot of errors with the "elevation profile" tool in Google earth for example.)

Nathan, slope angle shading is what I meant when I said gradient oops! ya it's a cool tool I had only used it looking at backcountry terrain but after posting here decided to check out vail... It's kind of interesting to look at the countours on china bowl and other in-bounds avy terrain. Also no bugs as far as I'm aware of, as long as USGS data is correct it should be good

if the distance is projected to the horizontal (such as measured on a map), use arctan.

if the distance is along the slope (such as published for lift lengths) use arc sin.

some claculators have dedicated buttons instead of using an inverse button. In addition, spreadsheets such as Excel have these functions in their function library.

Ghost and Montmorency have the correct math.  I use excel where the formula is:

=DEGREES(ATAN(F2/E2))

Where F2 is vertical and E2 is horizontal length.

I’m curious how people are measuring the vertical and horizontal values.  I have tried two methods both leveraging Google earth:

1.       Draw a line (path function at top of screen)

2.       In the properties box you can get the horizontal length in feet.

3.       Two options for vertical measurement:

a.       Zoom in to the endpoints, put your mouse on each endpoint and record the elevation of each.  Subtract these numbers and you have the vertical component.

b.      Zoom into the endpoints and view the value on a topo map overlay.  (national geographic has a free topo overlay based on the USGS survey)  Ideally with this method, you put the endpoints of the line right on the topo contour lines.

I’ve found such approaches to be fairly accurate as long as the vertical is 100’ or more.  The approach does have its limits though including (but not limited to):

+Google earth elevation data isn’t perfect.  Google keeps there method a secret but it appears to be based on USGS data and radar data from satellites.  The radar data gets thrown off by tall structures (buildings, trees, radio tower, etc)  All data appears to be averaged.  End result, the steeper slopes are likely to be undervalued and all measurements will suffer some error.

+USGA topo map data is more precise but it is only at 40’ intervals with the contour lines being low resolution.  Such limits introduce error, especially for shorter measurements.

+Google clearly has some errors in their data/tools.  Big ditches where there isn’t one, the summit appears downhill in Google, etc.