A Simpler Way to Find the Shaded Area?

[Saracen Rue]

TL;DR Summary

Is there an easier way to determine what percentage of this square is bound by the shaded area of 4 line segments

Consider the following scenario:

Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple?

I managed to determine that the purple shaded area is ##5\text{%}## as per my working below:

The only real problem I had with this is that it took me a genuinely long time to figure out. After I drew it up by hand, I suspected that line segment ##EG## was roughly equal to ##\frac{a}{3}## from visual inspection alone, and I did use a ruler to confirm this. However, it took me quite a while to prove it was mathematically.

I feel as though there must be a simpler way to go about solving the question using just geometry/trigonometry (I do realise that it'd probably be easy enough to solve if you put this on a Cartesian Plane with ##A## at the origin, but I wanted to try avoiding that method if possible), and if so, can anyone point me in the right direction?

 

[anuttarasammyak]

Say square ABCD=1
quadrilateral EFGH = ##\triangle## ACM - ##\triangle## AEF - quadrilateral HCMG
where
##\triangle##ACM=1/4,
##\triangle## AEF = ##\triangle## AEN - ##\triangle## AFN = 1/12 - 1/20,
quadrilateral HCMG = ##\triangle##CHD-##\triangle## GDM = 1/4 - 1/12.