How this calculator works
A wound roll is a spiral. The area of the side of the roll equals the length of the material multiplied by its thickness, because each turn adds a thin ring whose side area is length times thickness. The same side area is the difference between the area of the outer circle and the area of the core, that is pi/4 times the difference of the squares of the diameters. Setting the two equal and rearranging gives the length. The result assumes the layers sit tightly with no trapped air, which is why tape and film come out close and loose fabric less so.
Formula
length = pi x (outer diameter squared - core diameter squared) / (4 x thickness). Outer diameter = sqrt(core squared + 4 x thickness x length / pi). Thickness = pi x (outer squared - core squared) / (4 x length).
Worked example
A roll with a 25 cm outer diameter, an 8 cm core and 0.2 mm thick material holds pi x (0.0625 - 0.0064) / (4 x 0.0002) = 220.3 metres. Working the other way, 50 metres of 0.2 mm material on an 8 cm core needs an outer diameter of about 13.8 cm.
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Frequently asked questions
Why does the length grow with the square of the diameter?
Each new layer adds a ring of circumference pi times the diameter, and the diameters grow as you wind. Doubling the outer diameter roughly quadruples the material cross-section area, so the length grows much faster than the diameter.
How accurate is this for tape?
Very close, because tape winds tightly and has a uniform thickness. The main error is the thickness figure: a supplier may quote a nominal value that differs from the real caliper by a few percent.
Why does the result differ from the printed roll length?
Printed lengths are usually a minimum, and the material may be thicker at the ends. Tension during winding also stretches some films, so a measured roll can hold slightly more than the calculation.
What about a roll where the material is compressible?
Fabric, foam and paper compress, so the effective thickness is lower than the measured caliper and the real length is longer. Measure the length of one full turn and compare with the formula to find the effective thickness.