HomeCalculatorsMathSolid GeometryTorus Calculator
Math calculator

Torus Calculator

Ring torus V = 2π² R r² and A = 4π² R r with major R > tube r. Same engine as Solid Geometry. Not a sphere. Not horn or spindle. 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
Ring torus volume and surface from major R and tube r.
Scope
Ring torus R > r
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.

VolumeV = 2π² R r²
SurfaceA = 4π² R r
iSame engine as /solid-geometry. Default R=3, r=1 gives V=6π². Not a second engine.

How to use

1

Enter R and r

Major R must be greater than tube r. Both must be positive. Default R=3, r=1.

2

Read V and A

3-1 torus V = 6π², A = 12π².

Example calculations

Common configurations with formula and result.

ϟ

3-1 ring

R = 3, r = 1

V = 6π²
6π²
ϟ

Horn rejected

R = 1, r = 1

INVALID_INPUT
INVALID_INPUT

Torus calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A ring torus of major radius R>r and tube radius r>0 has volume 2π² R r² and surface 4π² R r. Horn (R=r) and spindle (R<r) tori are out of scope. This page is mode=torus on math.geometry.solid. Not /sphere. Not CAS.
What it calculates
Ring torus volume and surface from major R and tube r.
Inputs
  • R
  • r
Outputs
  • volume
  • surface
Formula
V = 2π² R r², A = 4π² R r
Assumptions
  • Ring torus R > r
  • Not horn or spindle
  • Not /sphere
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • R ≤ r → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
R=3 r=1 → V=6π²
Validation cases

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

  • R=3 r=1 → V=6π² A=12π²
  • R=1 r=1 → 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=torus on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Does this include a horn or spindle torus?

No. Only the ring torus with major R greater than tube r. Horn (R=r) and spindle (R<r) are out of scope.

Is this the sphere page?

No. The sphere stays on /calc/math/sphere. A torus is a tube of radius r whose centerline is a circle of radius R.

Where does this run?

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