I have a “flameless candle” (battery operated) with approximate dimensions 65 mm tall x 75 mm diameter (240 mm circumference).
I want to create a “living hinge” cylinder into which it will fit.
If I know that the base has a 75mm diameter, how can I calculate the length of the “living hinge” rectangle I want to cut so that the inside circumference is 240 mm and the material thickness is 3.1 mm?
Are there any online calculators for this kind of thing like the maker box apps out there?
A casual search for laser "living hinge" calculator shows AI has engulfed another design problem.
Picking the top one off the list, because why not:
More choices than I can handle in one afternoon! 
Make the wavy one 240 mm in the bendy direction and 65 mm in the stiff direction and it looks nice. You surely must tweak the edges to eliminate border cuts, but that’s in the nature of fine tuning.
Here’s a design I found awhile back. I have played with other hinges but keep coming back to this. maybe just scale it to what you want. gonna be some waste no doubt.
Round jewelry box – wood hinge - .120 birch copy.lbrn2 (911.0 KB)
I’d think, initially you should forget the idea of a thickness. The only time this will come into play is where the ends connect. If they are square, they won’t match the other end, at least they’d be visible.
You can use basic geometry to figure the size of the sheet you need, since you’re doing it flat. Just ensure you measurements are based on the inside.
mmmm, I’ll only get it for 235.5mm
but if you divide your 240 by Pi, you get around 67.5mm, to that you add 3mm material and you get Ø 70.5mm in inner dimension for your candle.
However, your “closing system” has not been taken into account, if it is more than 10mm you will have to compensate for it.
If your base is 75mm in (inner) diameter, you can calculate (75-3)*3.14= 226mm