c++ - Volume of irregular piece of pie -


imagine cut piece of pie known fixed height (h). cut in such way upper arc has angle a1 , bottom arc has angle a2. however, know upper arc has 2 radii (r1 , r2) , bottom arc has 2 different radii (r3 , r4). these 4 radii not connected anyhow each other (might elliptic axis, might total random). enter image description here technically, if not thinking pies, have 2 polygons (with same number of points), mass centers bit different each other. taking 2 points upper polygon, assume form local elliptic arc 2 radii , angle a1. (the other arc same polygon have different radii values , angle). can take 2 points lower polygon correspond 2 points of upper polygon (typically, using index values) , form lower local elliptic arc angle , radii values.

thus question how calculate volume of piece? again, 2 upper radii (r1 , r2), 2 bottom radii (r3 , r4), fixed height (h) , 2 angles (a1 , a2). note: centers of upper , bottom parts (essentially, polygons) might different.

thanks in advance help

this approximation might (assuming a1, a2 given in radians):

  • area of top "sector" a_top ~ (r1 + r2) * a1 / 4
  • area of bottom "sector" a_bot ~ (r3 + r4) * a2 / 4
  • volume of piece of pie v ~ (a_top + a_bot) * h / 2

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