Spherical Central Segment Calculator

Name: Spherical Central Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at a spherical central segment. This is formed by a sphere, of which two spherical caps are cut off. The circular slice planes don't need to be parallel, but must not cross within the sphere. The spherical central segment may extend beyond the sphere's center, unlike the calculator for the spherical segment, which uses the radii of the two caps and not their heights. The slice plane of the spherical central segment through the sphere's center and perpendicular to the cutting planes is a circular central segment.
Enter the radius of the sphere and the heights of the two cut off caps. Choose the number of decimal places, then click Calculate.

Formulas:

H = 2 r - h - i

a = \sqrt{h \cdot (2 r - h)}

b = \sqrt{i \cdot (2 r - i)}

L = 2 \cdot π \cdot r [2 r - (h + i)]

A = L + π \cdot (a^{2} + b^{2})

V = \frac{π}{3} \cdot [4 \cdot r^{3} - h^{2} \cdot (3 \cdot r - h) - i^{2} \cdot (3 \cdot r - i)]

pi:

π = 3.141592653589793...

Radiuses and heights have the same unit (e.g. meter), the areas have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

In the sketch, the cut planes of the spherical caps are parallel. However, this is irrelevant for the surface area and volume of this figure. As long as the two planes do not intersect, these values do not change simply because the cuts have different positions. If, however, the cuts are shifted relative to each other not just in one, but in two dimensions, so if there is no plane through the center of the sphere that intersects both cut planes perpendicularly, then the statement above about the circular central segment as the slice plane no longer applies.
The symmetry properties of the spherical central segment with parallel cuts are rotational symmetry about the axis through the centers of the cuts and mirror symmetry with respect to any plane in which this axis lies. If the two caps have the same height, the central segment of the sphere is even point-symmetric. These symmetry properties are lost if the cuts are shifted in one or in two dimensions.

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