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Spherical Segment Calculator

Calculate volume and surface area of a spherical segment (zone). A segment is the portion between two parallel planes.

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Sphere Radius (R)

Height h₁ (from top to first cut)

Height h₂ (from top to second cut)

Spherical Segment Diagram

Spherical Segment Formulas

Volume Formulas:

Volume: V = πh(3r₁² + 3r₂² + h²)/6
Alternative: V = Cap(h₂) - Cap(h₁)
Segment height: h = h₂ - h₁

Surface Area:

Curved area: A = 2πRh
Base radii: rᵢ = √(hᵢ(2R-hᵢ))
Total: SA = 2πRh + πr₁² + πr₂²

Understanding Spherical Segments

A spherical segment (or zone) is the portion of a sphere between two parallel planes. It has two circular bases of potentially different sizes.

Special Cases:

h₁ = 0: The segment includes the top cap of the sphere
h₂ = 2R: The segment includes the bottom cap
h₁ = 0, h₂ = R: Hemisphere
Through center: h₁ = R-d, h₂ = R+d for some d

Frequently Asked Questions

What's the difference from a spherical cap?

A cap is cut by one plane (one flat base), while a segment is cut by two parallel planes (two flat bases). A cap is a special segment where h₁ = 0.

Why is the curved area formula the same as for a cap?

Archimedes' theorem applies: the curved area between any two parallel planes equals 2πRh, where h is the distance between the planes. It only depends on R and h!

Where are spherical segments used?

Barrel-shaped containers, cylindrical tanks with domed ends, architectural elements, lens design, and calculating volumes of partially filled spherical tanks.

Related Calculators

Spherical Cap Sphere Hemisphere Frustum

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