parts of a ball

Let us consider in ${ℝ}^3$ a ball of radius $r$ and the sphere bounding the ball.

• Two parallel planes intersecting the ball separate between them from the ball a spherical segment, which can also be called a spherical frustum (see the frustum).  The curved surface of the spherical segment is the spherical zone.

• In the special case that one of the planes is a tangent plane of the sphere, the spherical segment is a spherical cap and the spherical zone is a spherical calotte.

• The lateral surface of a circular cone with its apex in the centre (http://planetmath.org/Sphere) of the ball divides the ball into two spherical sectors.

The distance $h$ of the two planes intersecting the ball be is called the height.

The volume of the spherical cap is obtained from

$$V=\pi h^2\left(r-\frac{h}{3}\right)$$

and the area of the corresponding spherical calotte and also a spherical zone from

$$A=\mathrm{\hspace{0.17em}2}\pi rh.$$

The volume of a spherical segment can be got as the difference of the volumes of two spherical caps.

The volume of a spherical sector may be calculated from

$$V=\frac{2}{3}\pi r^{2}h,$$

where $h$ is the height of the spherical cap of the spherical sector.