Volume of Ellipsoidal Cap

Date: 04/11/2001 at 11:19:54
From: R Shanks
Subject: prob309885_02molume of ellispoidal cap

I am doing research on cancer and need a way to properly determine the 
volume of tumors in lab animals. Such a tumor can best be described as 
an ellipsoidal cap. It is possible for me to measure the length 
width and height of the tumor, but I have looked in texts that contain 
common geometric formulas and I can't find the formula for the volume. 
Could you help? Thanks for your time.

Date: 04/11/2001 at 12:44:21
From: Doctor Rob
Subject: Re: Volume of ellispoidal cap

Thanks for writing to Ask Dr. Math.

The length, width, and height are insufficient to compute the volume.
Three parameters, a, b, and c, are necessary to specify the shape of
the ellipsoid, and another, r, to specify the plane of the base. You
have only three measurements, which would give you three equations in
the four parameters. These are insufficient to determine the values
of the parameters.

Set up a rectangular or Cartesian coordinate system with origin at
the center of the ellipsoid, and x-, y-, and z-axes each along one of 
the axes of the ellipsoid and so perpendicular to each other. Then the 
equation of the ellipsoid is

   x^2/a^2 + y^2/b^2 + z^2/c^2 = 1,  a, b, c > 0.

Let the cutting plane be x/a = r, -1 <= r <= 1, perpendicular to the
x-axis. Then the cap bounded by the plane and the surface of the
ellipsoid, with r <= x/a <= 1, has volume V, where

          a    b*sqrt(1-x^2/a^2)   c*sqrt(1-x^2/a^2-y^2/b^2)
   V = INT  INT                 INT                         dz dy dx,
         r*a  -b*sqrt(1-x^2/a^2)  -c*sqrt(1-x^2/a^2-y^2/b^2)

              a     b*sqrt(1-x^2/a^2)
     = 2*c*INT   INT                 sqrt(1-x^2/a^2-y^2/b^2) dy dx,
             r*a   -b*sqrt(1-x^2/a^2)

                   a
     = 4*Pi*b*c*INT  (1-x^2/a^2) dx,
                  r*a

   V = Pi*a*b*c*(1-r)^2*(2+r)/3,  -1 <= r <= 1.

Perhaps you can understand why this formula does not appear in the
standard collections.

The height is a*(1-r); the length and width are b*sqrt(1-r^2) and
c*sqrt(1-r^2). This is not enough to determine a, b, c, and r.
It is enough to specify the elliptical base, and the vertex of the
ellipsoid. You need another measurement, such as the length or
width at half the height, to get enough information to solve for a,
b, c, and r, and then compute the volume.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/