Spherical segment

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Spherical segment

A spherical segment is the solid you get when you cut a sphere with two parallel planes. The part between the planes is the segment, and the curved surface between the two circular bases is called the spherical zone. A segment looks like a cap with its top chopped off, and it’s a type of spherical frustum.

If the sphere has radius R, the two cutting planes create circular bases of radii a and b, and the distance between the planes is h. The volume of the segment is

V = π h (3a^2 + 3b^2 + h^2) / 6.

A handy way to understand this is to think of the segment as the sum of two cylinders (radii a and b, height h/2) plus a sphere of radius h/2. That gives

V = (π h/2)(a^2 + b^2) + (π h^3)/6 = π h (3a^2 + 3b^2 + h^2) / 6.

Special case: if one base radius is zero (b = 0), the segment reduces to a spherical cap, which matches the familiar cap formulas in this limit.

The curved surface area of the spherical zone (the area of the segment’s outer surface, excluding the top and bottom bases) is

A = 2 π R h.

This shows the zone’s area depends only on the distance h between the planes (and the sphere’s radius R), not on where the planes are located along the sphere.