HomeCalculatorsMathSolid GeometrySpherical Lune Calculator
Math calculator

Spherical Lune Calculator

Spherical lune A = 2r²θ from sphere radius and dihedral angle. Same engine as Solid Geometry. Not /spherical-wedge. 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 lune area from sphere radius and dihedral angle in degrees.
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 — 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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.

AreaA = 2r²θ (θ in radians)
Hemisphereθ = 180° ⇒ A = 2πr²
Sphereθ = 360° ⇒ A = 4πr²
Wedge laterallune A = wedge lateral
iSame engine as /solid-geometry. The solid of the same dihedral stays /spherical-wedge. One cutting plane stays /spherical-cap. Two parallel planes stay /spherical-zone. The closed sphere stays /sphere. Not a second engine.

How to use

1

Enter r and θ

Sphere radius must be positive. Dihedral angle θ is in degrees, 0 < θ ≤ 360. Default r = 1, θ = 90°.

2

Read A

Unit 90° lune has A = π. θ = 180° is half the sphere surface. θ = 360° matches the sphere.

Example calculations

Common configurations with formula and result.

ϟ

Unit 90°

r = 1, θ = 90°

A = π
π
ϟ

Hemisphere

r = 1, θ = 180°

A = 2π
2π
ϟ

Full sphere

r = 1, θ = 360°

A = 4π
4π

Spherical Lune calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A spherical lune is the sphere surface between two half-planes through a diameter. With radius r and dihedral angle θ in degrees, the area is 2r²θ with θ in radians. θ=180° is a hemisphere surface. θ=360° recovers the sphere. This page is mode=spherical_lune on math.geometry.solid. Not /spherical-wedge. Not /spherical-cap. Not /spherical-zone. Not /sphere. Not /spherical-sector. Not CAS.
What it calculates
Spherical lune area from sphere radius and dihedral angle in degrees.
Inputs
  • r
  • theta
Outputs
  • area
  • surface
  • theta_rad
  • sphere
  • hemisphere
Formula
A = 2r²θ (θ in radians)
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 < θ ≤ 360°
  • θ = 180° is a valid hemisphere surface
  • θ = 360° is a valid sphere
  • Not a wedge page
  • Not a cap page
  • Not a zone page
  • Not a sphere page
  • Not /spherical-sector
  • Not CAS
Units
  • length (consistent units)
  • angle (degrees in, radians in the formula)
  • area (length²)
Boundary conditions
  • r ≤ 0 or θ ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ > 360° → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
r=1 θ=90 → A=π
Validation cases

6 published on this page · 118/118 tests · Production surface contract 5/5 · View evidence

  • r=1 theta=90 → A=π
  • r=1 theta=180 → A=2π hemisphere=true
  • r=1 theta=360 → A=4π sphere=true
  • r=0 theta=90 → VALUE_MUST_BE_POSITIVE
  • r=1 theta=0 → VALUE_MUST_BE_POSITIVE
  • r=1 theta=361 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.18.0

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_lune on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the spherical wedge calculator?

No. The solid bounded by two half-planes stays on /calc/math/spherical-wedge. This page is the curved surface 2r²θ only.

Is this the spherical cap calculator?

No. One cutting plane stays on /calc/math/spherical-cap.

Is this the spherical zone calculator?

No. Two parallel planes stay on /calc/math/spherical-zone.

Is this the sphere calculator?

No. The closed sphere stays on /calc/math/sphere. Angle θ = 360° is a valid lune that matches the sphere surface.

Is this a spherical sector?

No. A sector bounded by a cap and a cone from the center stays on /calc/math/spherical-sector.

Where does this run?

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