1. I am trying to 'bisect' and object. See attached photo.  I am trying to completly get rid of the pink part of the circle outside the green box.  I want to retain the ability edit the fill of the quarter circle.  
2. I have watched several tutorials on youtube and used the 'manual.'
3. I've been playing with the path, intersection, union, etc and just am not getting it. 
4. I created a drawing in Inkscape. 
5. Windows 10 64 bit
6. I just installed inkscape
7. Version 1.2
8. No specia hardware
9. No helper programs

Thank you!

You just can use Path->Intersection with a duplicate of the rectangle and circle - stack order doesn´t matter. Or draw a circle (Slice with 2 radii) Start=270°, End: 180°.

or for a quarter circle:

The slice tool worked great once I had fiddled with it!  Thanks so much.  

To remove the outside of the circle, I'd make a duplicate of the rectangle, ensure it is in front of the circle, then use Path>Intersection.

https://inkscape-manuals.readthedocs.io/en/latest/boolean-operations.html