Name: Richard
Who is asking: Other
Level of the question: All

Question: I am trying to find the volume of a cone that is not round but oval. I am sure there is a way to measure this, but I have not found anything yet that I am willing to live with.



There is a nice result in geometry which says that if you take a region in the plane which has an area, a point P not in the plane and form a "tent" by joining every point on the boundary of the region to the P then the volume of the tent is

  1/3  (the area of the region at the base) (the height of the tent)

Your oval-based cone fits this description,

and hence if you know the height you only need find the area of the oval.

If the oval is an ellipse then it has a major axis and a minor axis. Let R be half the length of the major axis and let r be half the length of the minor axis,

then the area of the ellipse is

r R

Hence if the base of your cone is an ellipse then the volume is

  1/3  r R h