If you know the dimensions of a carpet, what will the dimensions be when you roll it up into a cylinder?

If you know the dimensions of a rolled-up carpet, what will the dimensions be when you unroll it?

This post answers both questions.

## Flexible carpet: solid cylinder

The edge of a rolled-up carpet can be described as an Archimedian spiral. In polar coordinates, this spiral has the equation

*r* = *h*Î¸ / 2Ï€

where *h* is the thickness of the carpet. The previous post gave an exact formula for the length *L* of the spiral, given the maximum value of Î¸ which we denoted *T*. It also gave a simple approximation that is quite accurate when *T* is large, namely

*L* = *hT*Â² / 4Ï€

If *r*_{1} is the radius of the carpet as a rolled up cylinder,Â *r*_{1} = *hT* / 2Ï€ and so *T* = 2Ï€ *r*_{1} / *h*. So when we **unroll** the carpet

*L* = *hT*Â² / 4Ï€ = Ï€*r*_{1}Â² / *h*.

Now suppose we know the length *L* and want to find the radius *r* when we **roll up** the carpet.

*T* = âˆš(*hL*/Ï€).

## Stiff carpet: hollow cylinder

The discussion so far has assumed that the spiral starts from the origin, i.e. the carpet is rolled up so tightly that thereâ€™s no hole in the middle. This may be a reasonable assumption for a very flexible carpet. But if the carpet is stiff, the rolled up carpet will not be a solid cylinder but a cylinder with a cylindrical hole in the middle.

In the case of a hollow cylinder, there is an inner radius *r*_{0} and an outer radius *r*_{1}. This means Î¸ runs from *T*_{0} = 2Ï€ *r*_{0}/*h* to *T*_{1} = 2Ï€*r*_{1}/*h*.

To find the length of the spiral running from *T*_{0Â } to *T*_{1} we find the length of a spiral that runs from 0 to *T*_{1} and subtract the length of a spiral from 0 to *T*_{0}

*L* = Ï€*r*_{1}Â² / *h* âˆ’ Ï€*r*_{0}Â² / *h* = Ï€(*r*_{1}Â² âˆ’ *r*_{0}Â²)/*h*.

This approximation is even better because the approximation is least accurate for small *T*, and weâ€™ve subtracted off that part.

Now letâ€™s go the other way around and find the outer radius *r*_{1} when we know the length *L*. We also need to know the inner radius *r*_{0}. So suppose we are wrapping the carpet around a cardboard tube of radius *r*_{0}. Then

*r*_{1} = âˆš(*r*_{0}Â² + *hL*/Ï€).