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When hiking up a steep hill without a trail, how do you determine the angle to switch back and forth?

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On trails that go up steep hills, there are usually switchbacks where the trail goes first one way and then other, resulting in an overall less steep path.

In the complete absence of a trail, how would I determine when I should switch back to the other direction and the optimum angle at which to go uphill?

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It's... basically completely up to you.

As with lots of things, it's about striking a balance - you can walk directly uphill if you like and it'll make for a shorter route, but it'll be hard going. You can also walk exactly following the countours of the hill - easy going, but you're not going to get anywhere very fast. Experiment, work out angle you're comfortable walking at.

You can also take into account a few things about your environment. If you're on wild land, there may be existing animal trails that you can follow, which might be easier walking. If there are man-made features like fences or walls, likewise - following them is often an easier walk. Also bear in mind your terrain - if you're on rocky ground or scree slopes, you'll probably want to take a shallower angle than if you're on compacted dirt.

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