I'm going to start mountaineering. Is there a recommended size per height/weight for ice axes for general mountaineering?
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With your hand on top of the head of the axe and holding it down by your side, the spike should come down to about your ankle. This will feel quite short, but when walking on steep ground (which you should be when taking an axe!), holding the axe in your uphill hand it will be a very useful length. Shorter is also lighter...
Ideally try and borrow one and see how you get on.
Some people prefer to have one long enough to use as a short walking stick. Others prefer to save weight and go as short as possible. In any case, it should be long enough so that you can use it properly to self-arrest. Everything else is optional.
You could rent a couple of ice axes of different sizes, go out to a steep snowy slope, and practice self-arrests to see what you're comfortable with. The practice will be valuable in itself. Keep in mind that stopping in powder, ice, and deep slush are all different experiences.
This is sourced from REI's How To Choose An Ice Axe.
As Chris mentioned, the axe should barely touch the ground when standing upright and with arms at the side. A rough guide to ice axe length is:
Too short is generally better than too long.
As AA Grapsas commented, it also depends on the type of climbing. Axes less than 60cm are generally for technical ice climbing and are best used for very steep or vertical slopes. Since they're shorter, they don't offer much leverage and are therefore bad for self arrest.
On the same note, axes over 70cm are generally too long for technical climbing, but better used for flat or lower angle slopes.
Mountaineering ice axes serve a few simple functions: self-arrest, belay, and T-anchor. (And occasionally cutting steps, but hopefully by the time you're doing that, you will already know what you like.)
For belay and T-anchors, any length of axe is fine.
For self-arrest, however, there are some factors you could consider:
So in summary, I find that (regardless of walking-stick usefulness) the best shaft length is the one between too long that it can impale my thighs, and too short that it can impale my abdomen. I generally measure this length as the diagonal from one shoulder to the hip bone on the opposite side, but you might find that this measurement varies for you.
To add a data point to the curve, I'm 6'3", and after much deliberation bought a 65cm DMM Cirque as a general winter hillwalking axe.
While I'm generally very happy with it, I'd definitely switch to a 60cm one next time. The only place where I find the extra length useful is cutting steps in descent.
Self-arrests with it are fine, but a shorter axe would be more wieldy for them.