I have, in metal, cut a $1.5''$ radius hemisphere (left half); now I want to cut a $1.8''$ diameter cylinder at $120^\circ$ from the horizontal and $0.333''$ deep on one edge and $0.031''$ deep at the other (referred to the edge of the hemisphere), with its' axis crossing the horizontal reference $0.226''$ to the left of the center of the hemisphere. I need to be able to calculate the amount of metal removed.
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Construct the hemisphere and the cylinder in any decent 3D CAD system (even a free one), and intersect them. Then ask the CAD system to calculate the volume of the intersection. Like MvG, I can't really understand what shape you're describing, so I can't do the construction myself.