Given the greatest angle and the circle radius I need to calculate the length of A segment.
The circle is tangent to both lines.
How can I do?
Let's call the centre of the smaller circle O, the point where the small circle touches the semi circle on the horizontal A, the other point which touches the semicircle B and the centre of the semicircle X.
Angle AXB = 116 degrees.
Angle OAX = angle OBX = 90 degrees.
Find angle BOA.
Once found, use the formula for the length of arc AB, that is $\displaystyle s = r\theta$
Assuming that you are talking about this arc AB.
A segment is an area. (Circular segment - Wikipedia, the free encyclopedia)
Probably all those lines could confuse.
I have 2 lines, S1 and S2, with a given angle beetween them (in this case is 116 degrees), then I have a circle tangent to both the lines with a given radius (in this case 20).
I need to find a way to calculate the length of the segment AB.
Oh, that's what you mean. Ok, no problem.
You now can find the angle AOB from your current diagram as I told you earlier. Well, let's call the point where the circle touches the other tangent X.
You know that angle XAB = 116 degrees.
That means angle XOB = 180 - 116 = 64 degrees.
(or [360 - 90 - 90 - 116] since both tangents are 90 degrees)
From there, the angle AOB becomes 64/2 = 32.
From there, you use the trigonometrical ratio tan.
tan 32 = Opp/Adj = AB/20
Solve for AB.