The length l of the Christmas lights is the hypothenuse of a rectangular triangle of height h, the height of the column. So, if the slope angle is α, we have sin(α) = h/l, or α = arcsin(h/l).

Soundness check: that doesn’t have a solution if h > l. Looks good.

does this assume the xmas tree is shaped like a column or a cone?

edit: ah, i re-read the original problem and it does mention column. i thought it was a xmas tree that was being wrapped.

That would be harder, yes. Reading https://en.wikipedia.org/wiki/Conical_spiral#Slope, you want a logarithmic spiral (you need a constant angle to make the problem make sense)

Luckily, arc length isn’t too gnarly for those (same Wikipedia page), but you still have one equation with two variables.

I would have to think hard about whether those give you a unique solution.

I also doubt that spiral would give you uniform coverage of the cone (and that probably, is the real requirement, not constant angles), but again, I would have to do some thinking.

oh, interesting variation for uniform coverage! that is indeed what i'd want for the tree. in building a road around a cone, a constant angle would be more desirable.

Suppose you’ve got a 16 foot strand of lights and an 8 foot column. If you unwrap the column in your mind, you can see you’ve got a right triangle with a hypotenuse of 16 and vertical leg of 8. What’s the angle that the hypotenuse makes with the floor? It’s the angle whose sine is opposite/hypotenuse = 8/16 = 1/2. That’s 30 degrees. So wrap the lights around the column at a 30 degree angle and it’ll be close (with a bit of slop thanks to rounding corners on the column).

my xmas isn't shaped like a column, it's a cone.

edit: ahh, the original question was for a column. i misread it and thought it was for a xmas tree.

If you unwrap a cone you get a circular sector. Similar idea.