/ calculating distance over hills

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buzby - on 20 Apr 2012
considering the years ive spent in the hills im a bit embarrased to ask this but here goes.
how to you calculate distance to travel from a to b when hills or mountains are in between for example.
if the distance as the crow flies between two points on a map was 10 miles but two mountains were in between at say around 4000 feet and you walked over both the total distance travelled is going to be more than 10 miles.
i appreciate you can take account of the fact that the route would not be in a straight line but how do you take in to account gains and drops in height.
I know im going to get the mick taken but id like to know anyway cheers.
Alasdair Fulton - on 20 Apr 2012
In reply to buzby: 10.44 Miles (Assuming you start and finish at sea level and also hit sea level when you drop back down between the two). I.e. really not worth worrying about.

If you were walking up and down something the steepness of a black run ~ 30degrees - I.e. 1 mountain, 30 degrees up and 30 down - it would make more difference, but even then it would only be +15.4% (11.54 miles for a 10 miles "as the crow flies").

I'm sure there must be some form of "rule of thumb" that a helpful MIA will come along and inform us of soon.

malky_c - on 20 Apr 2012
In reply to buzby: Take a wild guess? Works for me about 50% ot the time.

The easiest way is to use one of the many online route planners. I use this one: . That calculates distance and height gain, then estimates your time using Naismith's rule (you need to log in for it to do the height gain and time calculation).

Or you can measure out and count contours on a paper map and apply Naismith's rule manually:

"Allow 1 hour for every 3 miles (5 km) forward, plus 1 hour for every 2000 feet (600 metres) of ascent."

Depends how fast you walk, of course. I tend to be a little faster than Naismith, but not on really rough ground. Worth comparing the times it gives to some routes you have done to see if you need to make an adjustment.
In reply to malky_c: Or you could use our one: ;-)

But i agree, a guess usually does the trick. There's probably some mathematical rule re angles or whatnot but that sort of thing's way beyond me and really not necessary.
SouthernSteve on 20 Apr 2012
In reply to malky_c:
> (In reply to buzby) Take a wild guess? Works for me about 50% ot the time.

> "Allow 1 hour for every 3 miles (5 km) forward, plus 1 hour for every 2000 feet (600 metres) of ascent."

Consider the alternative if you are quite strong going up: "Allow 1 hour for every 4 km forward plus half an hour for every 600m climbed"

I used the first for years, but was recently Hannah'ed at GL and this seems to fit better with my pace. I take it as a personal affront to be slower than Mr Naismith!

Stuart (aka brt) - on 20 Apr 2012
In reply to buzby:

I guess if you know all the angles of slope and height gained, then you could throw the numbers into some trig equation couldn't you?

Definitely not a mathematician .
Anonymous on 20 Apr 2012 -
In reply to buzby:

Im firmly in the take a wild stab at the answer camp though usually I am pretty good at looking at a distance and guessing 20 minutes or about an hour or I'm gona miss tea tonight etc and usually get it there or there abouts!!

Naismiths rule doesn't work for me I can never make it work. Everytime I go out I seem to walk at a different pace. I have finished routes in 3 hours when naismtih says it should take 5 and then again I have taken 6 or 7 hours for the same route even when I thought I was travelling quite fast.

I'm sure you will find an answer or a method that works for you.
Rigid Raider - on 21 Apr 2012
In reply to buzby:

This obviously doesn't take account of the hills but a neat way of estimating distance is to follow the route on the map counting EVERY time it crosses a blue grid line, even if it only dives across briefly then back again. Halve the number and that's your distance in miles. There is a mathematical reason why it works but I have no idea how to explain it. It is amazingly accurate and more accurate the further you go.
almost sane on 21 Apr 2012 -
In reply to buzby:

The answer depends upon what you mean by "distance travelled."

For most purposes, when people talk about "distance travelled" on the ground they mean the horizontal component, ie how far it would be if the surface was completely flat. Then folks add a different measure which is the height gained and lost. So you have two numbers.

You appear to be asking a different question: if you were to lay a string over the route you travelled, then how long would that string be?
This is a complex question.
If you take your example of 10 miles horizontal distance incorporating two 4,000 feet hills, then there are two factors that will make the bit of string longer or shorter.
First, what is the total height gained and lost? The bigger this answer, the longer the string. This is obvious when you think about it: if you start roughly 2,000 feet (eg the Cairngorm car park) then your string will cover more ground than if you start at 200 feet (eg Glen Nevis). Likewise if the drop between mountains is only 500 feet your string will cover less ground than if the drop between hills is over 1,000 feet.

To add to the complexity, gaining 4,000 feet in a steady incline over 2 miles means the length of the string following your route will be shorter than that following a path that is level for one mile and then gains 4,000 feet in the second mile.

Finally you have the fractal problem of the earth's surface.
What if the route goes over peat hags and you are constantly going up and down slightly? Do you measure the distance covered differetly from walking along tussocky grass, differently from walking over smooth grass? If you walk for a half a mile horizontal distance over a horizontal boulder field, does your distance walked increase as the boulders get bigger?
kilner on 21 Apr 2012
If you want to calculate the exact distance you need to know the angle of the slopes.

This crude table shows distances and angles. (metric)

If you climb 1000m up a 90dg (vertical cliff) you gain no horizontal distance.

Climb 1000m at 85 dg and you travel 1004m but will measure only 87m horizontally on the map

Angle of slope - Distance travelled - horizontal distance measured on map
height gain = 1000m
90dg - 1000m - 0m
85dg - 1004m - 87m
80dg - 1015m - 176m
75dg - 1035m - 268m
70dg - 1064m - 364m
65dg - 1103m - 466m
60dg - 1155m - 577m
55dg - 1221m - 700m
50dg - 1305m - 839m
45dg - 1414m - 1000m

for slopes less than 45dg work on 1000m travelled horizontally
Angle of slope - Distance travelled - Height gain measured on map
40dg - 1305m - 839m
35dg - 1221m - 700m
30dg - 1155m - 577m
25dg - 1103m - 466m
20dg - 1064m - 364m
15dg - 1035m - 268m
10dg - 1015m - 176m
5dg - 1004m - 87m
0dg - 1000m - 0m

Hope this makes sense its a weird one to explain. i draughted this out on auto cad but you could use trigonometary to calculate exact distances.
pacman on 21 Apr 2012
In reply to buzby:

You don't. Assuming that what you really want to know is the time it'll take you to get from point A to point B you don't actually need to know the difference between the horizontal and slope distance.

Instead, work out your time estimate as follows.

Look at the ground to be covered between point A and point B. If your route is basically flat or on relatively gentle slopes use a guideline of X kilometres/miles per hour (depending on age, ground conditions, fitness etc.) to work out your time estimate. If your route involves steep ascent or descent then forget about the distance and just use a guideline of X metres/feet of ascent/descent per minute/hour (depending again on age, ground conditions, fitness etc.) to work out your time estimate.

If it's not obvious whether to use the flat ground speed or steep ground height gain/loss method then work it out both ways and you should find that your time estimates are about the same anyway.

Far simpler than Naismith's and it works.

Hope that makes sense and helps. If not see Pete Cliff's book Mountain Navigation book, he explains it much more clearly.
JLS on 21 Apr 2012
In reply to buzby:

I think you'd get a good approximation by using the following...

z = sqr root (x^2 + y^2)

z = actual distance in km

x = length of route on plan (crow flies distance) in km

y = number of 10m contours crossed (by the route) times 0.01km
Yrmenlaf on 21 Apr 2012 -
In reply to buzby:

2½ miles an hour over rough mountainous terain serves me well enough in the UK


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