In the picture above, the cross-hatched area is the area through which product can flow, if the knife gate is open to a width of 1.125"

I need to know cross-sectional area and perimeter (arc length a + arc length b) of the cross-hatched area.

Currently I am getting this data from AutoCAD. AutoCAD tells me the arc lengths and the area of the cross-hatch, so I move the knife in AutoCAD 1/8" at a time and record the arc lengths and area into a look-up table for a piece-wise linear scaling operation.

This is very tedious. I will continue the drudgery but I would much prefer a formulaic solution.

I have googled all of the things, and the results I get are much more complicated than I want them to be, and are exclusively solving for the area of two circles like this:

Not the crescent shape I need. And also not giving arc lengths/perimeter.

Is there any better way than what I'm already doing?