Spherical cap

A spherical segment or a spherical segment is a part of a spherical body, which is formed by cutting with a plane. A spherical segment in the form of a dome and having a surface area of a circular disk. A hemisphere is a special case of a spherical segment, in which the sectional plane containing the center of the sphere. The curved part of the surface of a spherical segment is spherical cup, also called a spherical cap or spherical cap.

Definitions

If you cut a sphere with radius with a plane, so the cut surface is a circular disk with radius. The two resulting from the cut part of the body of the hot ball ball ball sections or segments. A spherical segment has a dome shape with the height of the circular disc as the base. Contains the sectional area of the center of the sphere, then the names of the two resulting spherical segments hemispheres. With them is true.

The curved part of the surface of a spherical segment is called spherical segment, spherical cap or spherical cap.

Formulas

Spherical segment

Is the volume of a spherical segment

Accordingly, the surface of a spherical segment results ( for the derivation see ball zone) to

Spherical cap

The area of a spherical cap (excluding the area of ​​the final disc )

With

And thus

Follows

This formula summarizes Johannes Kepler in the words " The surface of a spherical segment is a circular area equal to, is the radius of the distance of the pole from one point of the base circle. " It is also true

Wherein the opening angle ( central angle ) is, from which

Followed. A spherical cap is thus determined by two variables, from which we can derive the other quantities:

As a spherical segment is a section of a spherical surface, the Gaussian curvature of the surface at all points is positive and equal.