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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 Engine tested · Specification checked · 111/111 tests · Production surface contract 3/3 · v1.18.0
Input interpretation
Enter values to calculate.
Result
—
Model
Spherical lune area from sphere radius and dihedral angle in degrees.
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Embedded snapshot: unpublished · Build schema 1.18.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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
  • 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 · 111/111 tests · Production surface contract 3/3 · 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: 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_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.