We must find the lengt of a cylinder role of fabric, which have 1000 windigs, diameter 10cm

The perimeter of the roll is pi times its diameter, or 31.4 cm. If you go around the perimeter 1000 times that equals a length of 1000 x 31.4 cm = 314 meters.

This calculation ignores the fact that the fabric has thickness, so as more fabric is would around the roll it takes more material for each winding due to the build up of material on the roll. If the thickness of the material is t, then each winding increases the overall diameter of the roll plus previous windings by 2t. The total length for 1000 windings is then:


L = 1000 \times \pi \times D + 2t \sum_{i=0} ^{999} i = 314m + 499,500 t


where the thickness t is given in meters