working out the slope angle the hard way

Sun, Dec 6, 2009

I was idly browsing January’s TGO, yes it’s only December but the mag is a month ahead and alongside the usual excellent content, there was an article on winter navigation. Now, as I’m heading for Winter ML assessment next winter (consolidation this winter), I took note of some points, especially the bit where it said that 6 index contours in 1cm on a 1:50,000 map is a slope angle of 30deg. That’s the most common angle for big avalanches, so it helps to be able to spot one on the map when you’re planning your day out in the winter mountains. Where does that number come from though? Well, with the rain lashing the windows and the gale rattling the slates (bit wild at the eBothy today), I thought I’d brush up some ancient maths skills to find out.

It’s a three step process. First you need a slope. So I used the compass to wander around the northern Cairngorms looking for a likely candidate and found a cracker on Bynack Mor:

Using the scale, 6 index contours fit perfectly onto the slope coming down from point 818m to the westwards bend in the burn in Strath Nethy. That’s the slope that bounds the northern side of  Allt a’Choire Dhuibh. It goes from the 500m contour just to the east of the bend, up to the 750m contour, after which the ground eases off considerably. You can see the main ascent path two burns north of the slope under scrutiny. Here’s the slope in more detail:

Next you need some data, so using the compass down the fall line gave a horizontal distance of 450m from the 500m to the 750m index contours.

Finally, you just play around with the numbers. We have the height gain and the horizontal distance. What we need now is the actual distance and from all those bits ’n bobs, we can work out the slope angle using trigonometry. It’s fun on a wet day, honest! Here’s the detail. First using Pythagoras to find the actual distance up the slope (marked “b” on the diagram) and the required angle (greek letter “alpha”):

Everyone knows Pythagoras, “a squared equals b squared plus c squared” and that gives the actual distance as 514.8m. I assumed 450.0m and 250.0m due to the accuracy of the map, so that’s four significant figures, hence an answer of 514.8m. The final part is a bit obscure as it uses the relation sinA = a/b which is basic trigonometry. As you can see, the answer is a cool 29.05 degrees, to four significant figures.

Whereas the compass scale can give you the rough slope angle:

• 1,50:000 - 6 index contours in 1cm = 30deg slope angle
• 1,25:000 - 3 index contours in 1cm = 30deg slope angle
• 1,40:000 - 4 index contours in 1cm (or 3mm between index contours) = 30deg slope angle

some easy maths also gives you a rough rule of thumb for the distance up the slope. If it's 30deg then you'll be walking about twice the height gained. i.e. for a 250m high slope, you'll be walking just over 500m.

Interestingly, I looked at the 1:40,000 Mountain Map of the slope and applying the 3mm rule above, the 30deg slope eases off a bit at 675m, so it’s only about 3/4 of the slope that is 30deg. 30deg +/- a degree here or there is still more or less 30deg to a load of fresh snow just itching to slide off into the glen though.