Surface of revolution volume

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How to find volume of toroid with rounded ribs? The problem I've faced is what radius should I use when I find width of small piece before integrating it from 0 to 2pi. What distance to choose: to the diameter or circle edge, or should I find the line which separate semicircle on the even parts (A=B).(A=B)

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It is easier if you find the volume of the hole and subtract it from the volume of the sphere. To find the volume of the hole you need to points of intersection of the circle with the line y-h. The middle part of the hole is just a cylinder. For the corners have two integrals to evaluate.

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Here is the relationship between the height of the hole and the radius of sphere when the two volumes are equal. enter image description here