Following on from this question: How many calories does hiking burn?

So in the same journey my girlfriends tracking app said we'd travelled 20km. Now this was based on us travelling on the flat, but we hadn't we'd also travelled upwards about 1Km too.

So how far had we actually walked? I had a thought that this would be something to do with Pythagoras's theorem but that seemed too far.

So if we'd walked, say 10Km as the "crow flies" and climbed 1Km how far had we actually walked (roughly)?

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    First math question on TGO. And everybody is like "YEAAAA" :) – Wills Aug 27 '14 at 20:04

Pythagoras is actually exactly what you would use, approximated as finely as you need for accuracy.

What I mean by approximated, is:

  • If you are following a continuous incline, you really only need one right angled triangle to calculate your hypotenuse, but if your incline varies, a more accurate figure will be gained by taking each change of incline as a new triangle. This also copes with you walking up and down slopes.

This gets complicated and annoying very fast, so for most purposes, you can approximate to 'a bit over' a single right angled triangle.

My preferred solution:

  • A GPS which either includes a height measurement (from GPS or barometric pressure) or orographic data in its built-in maps so it can calculate total distance traveled for you.

And in answer to your specific question, 10km across and 1km up gives you a total distance of 10049m (which is basically 10km :-)

  • My personal experience is that a handheld GPS device estimate on vertical elevation travelled is not accurate. Even when travelling on perfectly flat it will claim significant incline due to measurement uncertainty and pressure variations, and there's no way to distinguish that from actual elevation changes. – gerrit Aug 27 '14 at 18:35
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    Just depends on the quality of gps device, really:-) A good gps device can give you very accurate height data, and even for most ones, Altitude error is specified to be 1.5 x Horizontal error specification. This means that the user of standard consumer GPS receivers should consider +/-23meters (75ft) with a DOP of 1 for 95% confidence. – Rory Alsop Aug 27 '14 at 20:58

What exactly do you want to measure?

If you want to estimate shoe usage, it would be better to measure steps, not the distance.

If you want to estimate fatigue, than there's a heuristic, you should assume that 100m up is the equivalent of 1km on flat terrain.

So you have walked 20 km equivalents. It has taken you twice as much time as you would be walking on flat terrain, and you've burned about twice as many callories.

It's only the estimation, but from my personal experience, it's very accurate.

  • You should probably look at this question outdoors.stackexchange.com/questions/6574/…. I think this answer is more relevant there? – user2766 Aug 27 '14 at 13:00
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    100 metres up is 1KM of flat? I have never heard that before and we did near daily marches for the 10 years in the Army. I am not saying you are wrong, it just seems very vague. geokov.com/education/slope-gradient-topographic.aspx has a great explanation. For instance, a run of 247 metres over a rise of 280 metres is the equivalent to 373 metres. – Venture2099 Aug 29 '14 at 7:33
  • You can not properly estimate the effect of climbing if you do not include the horizontal distance. 100m over 1 km you notice, 100m over 10km is nothing. – Jan Doggen Oct 14 '18 at 18:47

You can get a good estimate of the distance walked by timing or pacing. Naismith's Rule (a way of estimating the time to walk a distance when ascents are involved) can help with the timing aspect but is only an estimation of the time taken to walk a certain distance taking ups and down into account. From the knowledge of expected average speed and time taken, you can estimate the distance travelled.

Pacing can be very accurate once practised but obviously is a bit onerous when you are trying to measure relatively long distances.

You can also get an indication of distance travelled when in hilly country by making use of the contour lines on maps.

See http://www.mcofs.org.uk/estimating-distance-travelled.asp for more details.


So if we'd walked, say 10Km as the "crow flies" and climbed 1Km how far had we actually walked (roughly)?

(looks like Math Markup isn't enabled here?)

km = sqrt( distance^2 + elevation^2 )

 = sqrt( 10^2 + 1^2 )

You added a whole 50 meters to your hike with that 1km elevation gain. That's assuming a steady slope.

If the route is up hill and down dale you get to repeat with every slope. Clearly not a job for a slide rule. You would need both an accurate GPS and an accurate terrain map

  • We don't have a lot of call for maths! – user2766 Aug 27 '14 at 14:05

The change in distance from flat is fairly small.

A 10% slope is about when you start toe striking instead of heel striking. Your leg is bent when your foot touches the ground and you have to straighten it. This is a lot more work. Your distance is about 0.5% extra due to the slope.

A standard stair has a rise over run of 7 over 10. Now you are going about 12% extra distance.

Typical scree slopes are at the angle of repose -- about 35-40 degrees. Now you are going about 25% extra distance.

This however is a bad way to look at it. Going up and down takes a lot more energy than traveling horizontally. When you are walking on slopes of 8% or lower, the biomechanics of your leg swing, and your stride length determine your speed. With some training you can lengthen your stride and increase your pace.

Around 10% slope you have to use a different motion, that is less efficient, much higher energy. For most of us at 20% slope (steep hill) conversation is short phrases separated by puffs. (I knew one lad who could sing going up a 15% slope. He also played the bag pipe. Another could carry on a conversation at slopes and speeds I could do little more than grunt. He had planted trees on mountainsides for 5 months.)

Coming down isn't a lot better. Your muscles are not designed for absorbing energy. Coming down hill you often need to take time to pick your way. A long descent can leave you at the bottom with "jello legs" quivering from the effort of contracting under load. This too you can train for, but a stair climber machine doesn't work. You have to go down slopes or stairs.

For running trips in one sense I don't care how far I really traveled. I want to have a good idea of what time it will take. So I have an estimater for the 'effective distance of elevation change'

In terms of planning distances I figure 5:1 for elevation. That is, a meter up effectively adds 5 meters horizontally. This is true for up and down both.

In practice the up part takes longer for any but the most fit, but coming down is still slower than flat (you are picking your foot landing more carefully.) Both going up and going down you are taking shorter steps.

So a 20 km loop route with a 1 km total of up and down will take about the same time, and leave the same fatigue more or less as a 25 km flat route.

In more detail: 8% is a magic number in terms of gradients. Up to about an 8% grade (8 foot climb per 100 foot horizontal) you can walk normally. You have a heel strike. Your step shortens up hill and lengthens down hill. At 10% for most people, you change to a toe strike, and you are climbing, rather than walking. This is not nearly as efficient. You will find that the vertical change entirely determines your speed. E.g. when you are walking you do 3 km/hour. When your are climbing you do 1200 meters an hour up. Your numbers will vary.

But the 5:1 rule is good enough for me.

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