I'm trying to find out the length of a loop which is enclosing 'n' number of cylinders.

I've found the length of the loop touching the cylinders, and now need to find the rest of the length of the loop (which is not touching the cylinders).

The formula I obtained for the length of the loop touching the cylinders is as follows;

L (of loop touching cylinders) = 4[2pi(r)(3pi/n)]

given that a pentagon has an internal angle some of 540 degrees.

I need clarification that this part of the formula is correct and need the remainder of the formula which I've stated above.

Here is the question straight from the assignment:

http://s5.tinypic.com/116oysj.jpg

Here is a top view of the diagram I've drawn, indicating the sections I've found the formula for:

http://i44.tinypic.com/mk95sn.jpg

Here is a question similar to mine, previously asked on physics forums (with 4 cylinders), which did not resolve with an answer;

https://www.physicsforums.com/attachment.php?attachmentid=14794&d=1216796703

Your help is greatly appreciated.