Geometry - Sperical Cap

Spherical Cap

Definition, Properties, and Formulae of a Spherical Cap

A spherical cap is the portion of a sphere cut off by a plane. It resembles a dome and is formed by slicing the top off a sphere. It's often encountered in geometry, architecture, and physics, particularly when analyzing volumes of partially filled spheres (like a bowl).

Spherical Cap

Key Parameters

  • \( R \): Radius of the sphere
  • \( h \): Height of the cap (from base to top of dome)
  • \( r \): Radius of the circular base of the cap
  • \( \pi \approx 3.1416 \)

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

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

Alternate form (in terms of sphere radius \(R\)):

\[ V = \pi h^2 \left(R - \frac{h}{3}\right) \]

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

\[ A = 2 \pi R h \]

Alternate expression (when \(r\) is known):

\[ A = \pi (r^2 + h^2) \]

Applications

  • Used in lens design and architecture (domes, tanks, vessels)
  • Appears in astronomy, meteorology, and modeling partial sphere coverage
  • Helps in determining volume of partially filled spherical containers
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