Spherical Segment Calculator

Name: Spherical Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at a spherical segment or spherical layer or zone of a hemisphere. Such a spherical segment is formed when a spherical cap smaller than a hemisphere is cut off by an even smaller, parallel spherical cap of the same sphere. A similar layer extending beyond the center of the sphere is the spherical central segment.
Enter the radiuses of the limiting circles and the height and choose the number of decimal places. Then click Calculate. i and j are the heights of the cropped parts of the hemisphere.

Formulas:

r = \sqrt{{a_{1}}^{2} + {[\frac{{a_{1}}^{2} - {a_{2}}^{2} - h^{2}}{2 h}]}^{2}}

i = \frac{{a_{1}}^{2} - {a_{2}}^{2} - h^{2}}{2 h}

j = r - h - i

L = 2 \cdot π \cdot r \cdot h

A = π \cdot (2 r h + {a_{1}}^{2} + {a_{2}}^{2})

V = \frac{π h}{6} \cdot (3 \cdot {a_{1}}^{2} + 3 \cdot {a_{2}}^{2} + h^{2})

pi:

π = 3.141592653589793...

Radiuses, height and distances 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.

The cross-section through a spherical segment is a circular layer.

The spherical segment can serve as an example for the application of integrals in spherical coordinates, particularly in determining volumes between two parallel planes that intersect a sphere. In differential calculus, the spherical segment is used as a model to calculate the centers of gravity or moments of inertia of spherical cuts. In architecture, domes or vaulted ceilings can be derived from spherical segments in a simplified manner to meet aesthetic or structural requirements.

The term layer is not always used in the sense described here. An atmospheric layer, for example, refers to phenomena in the form of a spherical shell. A spherical layer should therefore not be confused with a spherical shell. Both are parts of a sphere. However, the spherical layer, segment or spherical zone has two parallel, flat surfaces as its sides. The spherical shell or hollow sphere, on the other hand, consists of two equidistant, concentric spheres, which are curved surfaces, of which the smaller one is subtracted from the larger one.

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