Question

How to calculate angle of triangle when sides are known?

Answer 1

That's a very nice observation. If you have a three sided polygon (a triangle) and you know the side lengths then the figure is rigid and it should be possible to find the angles. If you have a polygon with more than three sides and you know the side lengths then the figure is not rigid and the angles can vary.

The relationship you are looking for is called the Law of Cosines. I can show you a proof at least for an acute triangle. If the triangle isn't acute you have to modify my proof slightly.
The triangle ABC has side lengths a, b and c. Draw AP perpendicular to BC, let h be the length of AP and x be the length of BP, then the length of PC is a - x.

Write Pythagoras' theorem for the two right triangles

x2 + h2 = c2
h2 + (a - x)2 = b2

I need one further equation

cos(C) = (a - x)/b or 
a - x = b cos(C)
and thus
x = a - b cos(C)

Expand the second equation to get

h2 + a2 - 2ax + x2 = b2

Using the first equation replace x2 + h2 by c2 then using the third equation replace x by a - b cos(C). Simplify to get

c2 = a2 + b2 - 2ab cos(C)

This is the Law of Cosines. If you know the side lengths a, b and c you can find cos(C) and hence the measure of the angle C.