HomeCalculatorsMathSolid GeometrySpherical Zone Calculator
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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 21/21 declared partitions (sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular-prism, regular-pyramid, regular-frustum, ellipsoid, spherical-cap, spherical-zone, spherical-wedge, spherical-lune, spherical-sector, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Spherical zone volume, recovered sphere radius, and surface from two base radii and height.
Scope
Euclidean 3-space
Verification
Engine tested · Source checked · v1.18.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.18.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-21.solid-cvp-all-modes
Verification revision
2026-09-21.solid-cvp-all-modes · 4/4 property · digest d81459afdb09
Legacy regression
118/118 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
26/26 golden · 27/27 CVP boundary · 32/32 invalid · 4/4 property · 8/8 cross-interface · 2/2 cross-calculator · 3/3 CVP contract · Manifest
Sources
Sources
Evidence
43 legacy golden · 25 legacy boundary · legacy regression suite · 26/26 oracle-backed golden · 32/32 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.18.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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
  • The Result Card includes an engine-linked 2-D generating section or silhouette of the published lengths. Not a 3-D mesh. Same pack as REST/MCP.
  • 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 · 118/118 tests · Production surface contract 5/5 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare) — §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area
    Supports: Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.18.0

Background

Interpretation and common distinctions.

What does this spherical zone calculator compute?

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.

The published formula or method is V = (πh/6)(3a² + 3b² + h²), A = 2πrh + πa² + πb². The following references support the stated mathematics or method:

  • Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare): Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Lengths, areas, and volumes have different units. Check whether a reported area is curved/lateral or total and use the shape-specific dimension constraints.

The worked example uses a = 1, b = 1, h = 1 with V = 7π/6. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.solid v1.18.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Cone Calculator, Cylinder Calculator, Dodecahedron Calculator, Ellipsoid Calculator, Frustum Calculator, Icosahedron Calculator, Octahedron Calculator, Prism Calculator . Explore all Solid Geometry.

→ 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. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dodecahedron Calculator Regular dodecahedron V = (1/4)(15+7√5)a³ and A = 3√(25+10√5) a² from one edge. Same engine as Solid Geometry. Not an icosahedron. Not an n-gon prism. Not CAS. Runs locally. › → Ellipsoid Calculator Ellipsoid V = 4/3 πabc and surface from the exact elliptic-integral model, numerically evaluated. a = b = c is a sphere. Same engine as Solid Geometry. Not /sphere. Not the 2-D ellipse. Not CAS. Runs locally. › → Frustum Calculator Right circular frustum V = (1/3)πh(R²+Rr+r²) from two radii and height. Same engine as Solid Geometry. Not /cone. Not ISO 5167-5. Not CAS. Runs locally. › → Icosahedron Calculator Regular icosahedron V = (5/12)(3+√5)a³ and A = 5√3 a² from one edge. Same engine as Solid Geometry. Not an octahedron. Not a dodecahedron. Not CAS. Runs locally. › → Octahedron Calculator Regular octahedron V = (√2/3)a³ and A = 2√3 a² from one edge. Same engine as Solid Geometry. Not a tetrahedron. Not an icosahedron. Not CAS. Runs locally. › → Prism Calculator Right rectangular prism V = abh from three edge lengths. Same engine as Solid Geometry. Not a frustum. Not a pyramid. Not CAS. Runs locally. ›

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.