Goal: calculate the length of a cylindrical roll of tape given the thickness of the tape, the internal and external diameter of the roll.
Firstly, I considered the roll of tape to be a bunch of concentric "circles" rather than a spiral.
Let
- $L=$ length of the tape
- $t=$ thickness of the tape
- $d-2t=$ internal diameter (diameter of the cardboard in the middle)
- $D=$ external diameter (diameter of the largest circle)
- $\alpha= \frac{D-(d-2t)}{2t}$ (the total number of circles of tape)
Then,
$$L=\sum_{i=0}^{\alpha-1} \pi(d+2it)$$
$$=\sum_{i=0}^{\alpha-1} \pi d + \sum_{i=1}^{\alpha-1} 2\pi it$$
$$=\alpha \pi d + \alpha (\alpha -1) \pi t$$
$$=\alpha \pi [d + (\alpha -1)t]$$
I have tested this formula for a few different values of $\alpha$, $d$ and $t$ and it seems to give the correct answer.
However, a roll of tape is not a bunch of concentric circles.
Is there any way to calculate the exact length of tape given these parameters?
Furthermore, how effective would using the formula above be?
Find the area of rolled tape and that would be equal to the area of wounded up tape.