Geometry - Sperical Segment

Spherical Segment

Definition, Properties, and Formulae of a Spherical Segment

A spherical segment is a three-dimensional shape formed by cutting a sphere with two parallel planes. It consists of the part of a sphere between these two planes and has two circular faces—one on the top and one on the bottom.

Spherical Segment

Key Parameters

  • \( R \): Radius of the sphere
  • \( h \): Height of the segment (distance between the two cutting planes)
  • \( r_1, r_2 \): Radii of the top and bottom circular faces
  • \( \pi \approx 3.1416 \)

1. Volume of the Spherical Segment \(V\)

\[ V = \frac{1}{6} \pi h \left( 3r_1^2 + 3r_2^2 + h^2 \right) \]

Alternate form (simplified version):

\[ V = \frac{1}{6} \pi h^3 + \frac{1}{2} \pi (r_1^2 + r_2^2) h \]

2. Surface Area of the Spherical Segment \(A\)

\[ A = 2 \pi R h \]

Note: This includes only the curved surface area, not the areas of the two circular faces.

Applications

  • Used in calculating fluid volumes in spherical tanks cut at different levels
  • Appears in architectural domes and geometric modeling
  • Useful in astronomy for modeling spherical celestial regions
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