Make the circle as a shape.
Convert circle to a path.
Select two of the four control nodes, insert a node.
Inserted node will appear at midpoint on path between selected nodes.
Do not move any nodes yet, you'll damage path's circularity.
Repeat steps 3,4 recursively until you have a node close enough to your
desired cut point at each side of the segment to be removed.
Hmm, why can't you just drag the rightmost controls of the circle as shape, without converting to path? That's exactly what these controls are for. If you don't need the radii, drag the controls from inside the circle. Use Fill&Stroke dialog to remove fill if you don't need it. It's very easy to create an arc of any angle, with or without radii, using these controls. Dragging the controls with Ctrl snaps the angle.
- Selecting the node of the cut segment can be difficult, since they
share the same location as nodes on the preserved path. Use undo immediately if you select the wrong one. You definitely don't want to disturb nodes relating to the curvature of your circle.
In such cases I usually press an arrow key and see which node moves. If it's the wrong one, I move it back and press Tab or Shift+Tab to move selection to the next/previous one, and then try again. It's quite fast when you get used to it.
- Clean up extra nodes near the endpoints *before moving any*. I've found
that you can often remove some of the intermediate nodes placed halfway between other nodes. Removal of a node will only be practical if there is enough weight available in the remaining nodes to replicate the original curvature. Ride the undo button immediately if you remove a critical node and lose curvature.
Pressing Ctrl+L (simplify) will remove all unnecessary nodes at once with very small, if any, distortion in the shape.