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
Assuming that you are talking about this arc AB.
A segment is an area. (Circular segment - Wikipedia, the free encyclopedia)