9 unit circles are packed into a square. What is the length of longest path connecting two opposite corners of square?

Question

Nine unit circles are packed into a square, tangent to their neighbors and to the square. What is the length of the longest smooth path connecting two opposite corners of the square?

Assumptions:
- The path must be continuous and follow the lines in the diagram; that is, it must be made up of portions of either the circles or the outside square.
- The path may not change direction suddenly.
- The path may not contain any loops.
- The path may not touch or cross itself at any point.

Answer 1

From the bottom end to the top the distance between the corners can be expressed as a sum shown below
1+π+π+(π/2)+(3π/2)+π+(π/2)+(3π/2)+π+(π/2)+1 = (2+ 17(π/2)).