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Calculating areas between QGraphicsItems

Hi!
Currently I am trying to implement a bucket tool to fill with color the areas generated by the intersection of several QGraphicItems. Let me describe a couple of examples of my requirement.
Example 1: If I have three independent paths that form a shape like this:
Could it be possible to calculate the area created by them, so I can add fourth element with the yellow shape?Example 2: If I have two ellipses
Could it be possible to calculate the area generated by the intersection of both elements, so I can include a third element with the yellow form?I am guessing that I will have to implement several algorithms, one for every case. Any way, I just started my research about this topic, so any suggestion or reference related to this problem is very welcome!
Thanks.