sqrt(3)/6 = 1/3.46 which is slightly smaller than 1/PI (1/3.14).
There are no losses with both formulas. with sqrt(3)/6 the pixel count stays the same. with 1/PI the pixel count is slightly bigger. Both formulas are viable.
firefoxer: please be more respectful of other people's work and resources in the future. You have hot-linked the image of the formula from my blog without permission.