Spherical Sector Calculator
Spherical sector V = (2/3)πr²h from sphere radius and cap height. h = r is a hemisphere. h = 2r is a sphere. Same engine as Solid Geometry. Not /spherical-cap. Not /cone. Not /sphere. Not CAS. Runs locally.
Enter values to calculate.
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 sector volume, base radius, and surface from sphere radius and height.
- Scope
- Euclidean 3-space
- Verification
- Engine tested · Specification 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
- 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
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Evidence
- 43 legacy golden · 25 legacy boundary · legacy regression suite · 26/26 oracle-backed golden · 32/32 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Enter r and h
Sphere radius and sector height must be positive, with h ≤ 2r. Default r = 1, h = 1 (hemisphere).
Read V, a, and A
Unit hemisphere has V = 2π/3 and A = 3π. At h = 0.5 the sector is larger than the cap. h = 2 matches the unit sphere.
Example calculations
Common configurations with formula and result.
Unit hemisphere
r = 1, h = 1
Half height
r = 1, h = 0.5
Sphere identity
r = 1, h = 2
Spherical Sector calculator specification
Version 1.18.0 · Engine tested
- Engine tested 118/118 tests · Production surface contract 5/5
- Named expert review Not performed
- Calculation version 1.18.0
- Definition
- A spherical sector (spherical cone) is the solid of a spherical cap plus the cone from the sphere center to the cap base. With radius r and height h, 0<h≤2r, volume is (2/3)πr²h, base radius a=√(h(2r−h)), and surface πr(2h+a). h=r is a hemisphere. h=2r is the full sphere. This page is mode=spherical_sector on math.geometry.solid. Not /spherical-cap. Not /cone. Not /sphere. Not /spherical-wedge. Not /spherical-lune. Not CAS.
- What it calculates
- Spherical sector volume, base radius, and surface from sphere radius and height.
- Inputs
- r
- h
- Outputs
- volume
- surface
- lateral
- base
- a
- sphere
- hemisphere
- Formula
V = (2/3)πr²h, A = πr(2h + a)- 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.
- 0 < h ≤ 2r
- h = r is a valid hemisphere
- h = 2r is a valid sphere
- Not a cap page
- Not a cone page
- Not a sphere page
- Not a wedge page
- Not a lune page
- Not CAS
- Units
- length (consistent units)
- volume (length³)
- area (length²)
- Boundary conditions
- r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
- h > 2r → INVALID_INPUT
- unknown mode → INVALID_MODE
- Example
- r=1 h=1 → V=2π/3 A=3π
- Validation cases
5 published on this page · 118/118 tests · Production surface contract 5/5 · View evidence
- r=1 h=1 → V=2π/3 A=3π hemisphere=true
- r=1 h=0.5 → V=π/3
- r=1 h=2 → V=4π/3 A=4π sphere=true
- r=1 h=0 → VALUE_MUST_BE_POSITIVE
- r=1 h=3 → INVALID_INPUT
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.18.0
Related tools
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.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second solid engine?
No. It is mode=spherical_sector on math.geometry.solid, the same engine as /calc/math/solid-geometry.
Is this the spherical cap calculator?
No. The cap without the cone from the center stays on /calc/math/spherical-cap. At h = r both are a hemisphere. At other heights the sector is larger.
Is this the cone calculator?
No. The right circular cone stays on /calc/math/cone. This page is a cap plus the cone from the sphere center.
Is this the sphere calculator?
No. The closed sphere stays on /calc/math/sphere. Height h = 2r is a valid sector that matches the sphere volume and surface.
Is this a spherical wedge?
No. Two half-planes through a diameter stay on /calc/math/spherical-wedge.
Is this a spherical lune?
No. The curved surface 2r²θ stays on /calc/math/spherical-lune.
Where does this run?
Locally in the browser by default. REST and MCP call the same solid-geometry engine.