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Math calculator

Spherical Zone Calculator

Spherical zone V = (πh/6)(3a²+3b²+h²) from two parallel base radii and height. Same engine as Solid Geometry. Not /spherical-cap. Not /sphere. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 111/111 tests · Production surface contract 3/3 · v1.18.0
Input interpretation
Enter values to calculate.
Result
—
Model
Spherical zone volume, recovered sphere radius, and surface from two base radii and height.
Scope
Euclidean 3-space
Verification
Engine tested · 111/111 tests · Production surface contract 3/3 · Specification checked · v1.18.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
42 golden · 23 boundary · 46 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.18.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

VolumeV = (πh/6)(3a² + 3b² + h²)
Sphere radiusr = √(a² + ((b²−a²+h²)/(2h))²)
SurfaceA = 2πrh + πa² + πb²
Zone (lateral)2πrh
Symmetrica = b ⇒ r² = a² + (h/2)²
iSame engine as /solid-geometry. One cutting plane stays /spherical-cap. The closed sphere stays /sphere. Not a second engine.

How to use

1

Enter a, b, and h

Base radii and zone height must be positive. Default a = 1, b = 1, h = 1 (symmetric zone).

2

Read V, r, and A

Symmetric 1-1-1 has V = 7π/6. Unit-sphere band a = b = √3/2, h = 1 has V = 11π/12.

Example calculations

Common configurations with formula and result.

ϟ

Symmetric 1-1-1

a = 1, b = 1, h = 1

V = 7π/6
7π/6
ϟ

Unit-sphere band

a = √3/2, b = √3/2, h = 1

V = 11π/12
11π/12
ϟ

Unequal 2-1-1

a = 2, b = 1, h = 1

V = 8π/3
8π/3

Spherical Zone calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A spherical zone between two parallel planes, with base radii a, b and height h, has volume (πh/6)(3a²+3b²+h²), sphere radius r = √(a² + ((b²−a²+h²)/(2h))²), curved area 2πrh, and total surface 2πrh+πa²+πb². a=b is a symmetric zone. This page is mode=spherical_zone on math.geometry.solid. Not /spherical-cap. Not /sphere. Not a spherical wedge. Not CAS.
What it calculates
Spherical zone volume, recovered sphere radius, and surface from two base radii and height.
Inputs
  • a
  • b
  • h
Outputs
  • volume
  • surface
  • lateral
  • base
  • r
  • symmetric
Formula
V = (πh/6)(3a² + 3b² + h²), A = 2πrh + πa² + πb²
Assumptions
  • Euclidean 3-space
  • Two parallel planes
  • a > 0, b > 0, h > 0
  • a = b is a valid symmetric zone
  • Not a cap page
  • Not a sphere page
  • Not a spherical wedge
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=1 b=1 h=1 → V=7π/6
Validation cases

5 published on this page · 111/111 tests · Production surface contract 3/3 · View evidence

  • a=1 b=1 h=1 → V=7π/6 symmetric=true
  • a=√3/2 b=√3/2 h=1 → V=11π/12 r=1
  • a=2 b=1 h=1 → V=8π/3 r=√5
  • a=0 b=1 h=1 → VALUE_MUST_BE_POSITIVE
  • a=1 b=1 h=0 → VALUE_MUST_BE_POSITIVE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.18.0

Other calculators in this family: Angle Between Lines Calculator, Annulus Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

No. It is mode=spherical_zone on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the spherical cap calculator?

No. One cutting plane stays on /calc/math/spherical-cap. This page is two parallel planes.

Is this the sphere calculator?

No. The closed sphere stays on /calc/math/sphere.

Is this a spherical wedge?

No. A wedge bounded by two half-planes is /calc/math/spherical-wedge.

Where does this run?

Locally in the browser by default. REST and MCP call the same solid-geometry engine.