How Long is a Piece of Christmas Lighting?

The goal of this short paper is to detail how long a piece of Christmas lighting you need in order to decorate a Christmas tree.

The Model

The model of the Christmas tree and the lighting is very simple. It rests on eight assumptions and notations. For clarity, I'll use capital letters for constants or parameters to the problem, and lower-case letters for variables.
• The part of the Christmas tree you want to put lights onto is a cone.
• The lights will be strung on the outside of the Christmas tree.
• The height of the part of the tree that you want to decorate is H.
• The radius of the tree at the bottom of the part you want to decorate is R.
• The lights will be strung around the tree in such a way that at each circuit of the tree, the height of the line increases by the same amount (even though the distance from the centre line of the tree will vary).
• You want N circuits of the tree.
• The path of the wire can be parameterised by the number of radians wound around the centre of tree as the wire goes from the top of the cone to the bottom, and we'll usually symbolise this parameter by ✓. For N windings around the tree, ✓ varies from 0 at the top of the tree to 2pieN at the bottom.
• The length of lighting that you need to decorate the whole tree will be L. This is enough information to work out the answer. Let's get calculating. Because it's what I'm used to, I'll use metric in the examples. But only one measure is involved, namely distance, so you can just as easily use feet, yards, cords, or furlongs.