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I don't know how else to state my question. What I'm looking to do seems like it ought to be very simple.

I want to take a brand new torus, and cut it in half. i.e. the exact shape you'd get if you cut a real donut in half (with the sliced edges being a face (I don't care if there are multiple edges / faces).

The same thing, but only cutting out 1/4, i.e. removing the bits of the torus shown in the selected image, and sealing / adding a face to the cut edges.

enter image description here

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If I understood you correctly:

you may hit the P and select the Selection

It will detach the selection from your main object and create a new independent object.

After that you may select the edges and hit F

Something like this:

enter image description here

enter image description here

enter image description here

If its not your required result, please try to update the question with more details

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