Math CentralQuandaries & Queries


Question from Susan, a teacher:

How would I find the surface area of a lemon

Hi Susan,

The shape of a lemon is approximately an ellipsoid, in fact a prolate ellipsoid. Prolate just means that if you slice the ellipsoid as you would normally cut a lemon in half then the cross section is a circle, and if you cut it in half the other way the cross section is an ellipse with the short axis of the ellipse the same as the diameter of the circle you got in the other direction. The only reason I mention this is that there is an exact formula for the surface area of an oblate ellipsoid (and also a prolate ellipsoid) but for a general ellipsoid the surface area can't be written in terms of elementary functions. There is an approximation however of the surface area of a general ellipsoid that, in my opinion, is easier to use then the exact expression for a prolate ellipsoid.

I am sure this is more than you wanted to know. You can find the approximation in my response to an earlier question. Just take a and b to each be half the width of the lemon and c half the length. The exact expression can be found on Wikipedia. Just scroll down to the Surface Area section.

I hope this helps,


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