Problem 314

The Mouse on the Moon

The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a 500 m by 500 m square area, but they will possess only that area which they wall in. 251001 posts have been placed in a rectangular grid with 1 meter spacing. The wall must be a closed series of straight lines, each line running from post to post.

The bigger countries of course have built a 2000 m wall enclosing the entire 250 000 m^{2} area. The Duchy of Grand Fenwick,
has a tighter budget, and has asked you (their Royal Programmer) to
compute what shape would get best maximum enclosed-area/wall-length
ratio.

You have done some preliminary calculations on a sheet of paper.
For a 2000 meter wall enclosing the 250 000 m^{2} area the
enclosed-area/wall-length ratio is 125.

Although not allowed , but to get an idea if this is anything better:
if you place a circle inside the square area touching the four sides the
area will be equal to π*250^{2} m^{2} and the perimeter will be π*500 m, so the enclosed-area/wall-length ratio will also be 125.

However, if you cut off from the square four triangles with sides 75 m, 75 m and 752 m the total area becomes 238750 m^{2} and the perimeter becomes 1400+3002 m. So this gives an enclosed-area/wall-length ratio of 130.87, which is significantly better.

Find the maximum enclosed-area/wall-length ratio.

Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.

**
These problems are part of
Project Euler
and are licensed under
CC BY-NC-SA 2.0 UK
**