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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?

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    [Seems similar](http://math.stackexchange.com/questions/1633704/the-length-of-toilet-roll), no? – 2017-02-15
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    yes, thank you! – 2017-02-15

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Find the area of rolled tape and that would be equal to the area of wounded up tape. enter image description here

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    that is not what I am asking – 2017-02-15
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    @ConnorGaughan, But didn't you ask this: ...Is there any way to calculate the exact length of tape given these parameters? – 2017-02-15
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    I probably shouldn't have said "using these parameters" as the parameters don't make sense when we have a spiral instead of concentric circles, my bad (inner and outer diameters don't exist right?) – 2017-02-15
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    @ConnorGaughan, Since the thickness of an usual tape is very thin then this method is quite accurate, but when it comes to a noticeable thick material then we need a different way of calculation. – 2017-02-15
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    I have posted a similar question (about the thickness of a toilet paper) in my FB. I found your picture is much nicer than mine. Can I borrow it? – 2017-03-03
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    @Mick, Yes of course my friend, you can use it. – 2017-03-03