Welcome to Math Help Forum!
You've certainly set quite a poser here. I've been working on it with a spreadsheet and have come up with the following:
Originally Posted by vitchalhey
a = 7862There are inevitably slight errors in these measurements, mostly due to the approximations in the angles. These were only given to the nearest degree in your diagram; this was not accurate enough for the calculations. You can verify this by using them to calculate the diagonal lengths, each in two different ways, using the cosine rule on the triangles.
b = 7849
c = 5767
d = 8824
e = 8838
f = 5054
g = 16672
h = 7371
I have therefore adjusted the angles a little by using 1 or 2 decimal places to make the data more consistent. You will see that, to the nearest degree, the angles are the same as the originals.
I attach an Excel spreadsheet showing the detailed calculations, and an explanatory diagram showing the names of the various points (deliberately drawn to exaggerate the angles). I hope that you have Excel available to check the validity of the calculations. You will see that I have achieved a fair degree of accuracy in the areas A1 - A4.