Spherical segment

A spherical layer, also called spherical disc, is part of the full sphere. It has two parallel sectional surfaces. Intuitively, one can imagine the way that you expect from a ( assumed to be spherical) apple from the center cut out a slice.

Mathematical clarification

The spherical layer is (whether the radius ) from a solid sphere by two parallel, the ball really intersecting planes ( their distance is ) cut out.

To calculate a few terms are needed:

The larger of the two arising from the cut parallel circular surfaces is called base and designated by the letter G, is its radius. The smaller is called top surface, and referred to by the letter D, is its radius. The third of the delimiting surfaces, the outer surface is also called a spherical zone and denoted by M.

Formulas

Is the volume of the truncated cone, which is inscribed in a sphere layer, and the length of its generatrix, as is

Derivation of the equations

Surface area of ​​the spherical zone

The ball-shaped region is produced by the edge of the cross-sectional area to the y-axis is rotated. In this case, due to the rotational symmetry of the section of the peripheral surface is observed with an xy - plane, which intersects with the ball in the poles and at the center, as a function or intended. There are then the circumferences (where ), multiplied by the infinitesimal arc length, or added continuously integrated, which results in the desired area

Volume of the sphere layer

The spherical segment is generated by the cross -sectional area to the y-axis is rotated. Then for the volume