Spherical segment







In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.

The surface of the spherical segment (excluding the bases) is called spherical zone.



If the radius of the sphere is called $R$, the radii of the spherical segment bases are $a$ and $b$, and the height of the segment (the distance from one parallel plane to the other) called $h$, then the volume of the spherical segment is


 * $$V = \frac{\pi}{6} h \left(3 a^2 + 3 b^2 + h^2\right).$$

For the special case of the top plane being tangent to the sphere, we have $$b = 0$$ and the solid reduces to a spherical cap.

The equation above for volume of the spherical segment can be arranged to
 * $$V = \biggl [ \pi a^2 \left (\frac{h}{2} \biggr ) \right ] + \biggl [ \pi b^2 \left ( \frac{h}{2} \biggr ) \right ] + \biggl [ \frac{4}{3} \pi \left( \frac{h}{2} \right)^3 \biggr ]$$

Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius $a$ and the second of radius $b$ (both of height $$h/2$$) and a sphere of radius $$h/2$$.

The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by


 * $$A = 2 \pi R h.$$