Interesting take on problem

There is this nice problem of a cloth of 25 m long 0.2mm thick being wound around a cylinder and the diameter of the resulting cylinder cum cloth is 10 cm. Find the diameter of the cylinder? My approach was to consider the diameter of the cylinder to be d mm and each successive wrap to increase the diameter by .2 mm and you obtain a series with n and d as unknowns. A second equation can be had by realizing that 50 = d/2+n* .02. Now you have two equations and two unknowns and it's easy to solve with a compute.
The second method which was author's was quite ingenious. It was something finding the difference between the area of the cloth plus cylinder - the area of the cylinder = area of the cloth. This at first instance didn't seem like a viable option to me. But when you give it a little thought it seems totally plausible and this is much quicker. Given the numbers in this question we have
Pi*10^2/4-Pi*D^2/4=2500*.01
Pi*25-Pi*D^2/4=25
25(Pi-1)/=Pi*D^2/4
100(Pi-1)/Pi=D^2
D=8.6