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)