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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 Engine tested · Specification checked · 111/111 tests · Production surface contract 3/3 · v1.18.0
Input interpretation
Enter values to calculate.
Result
—
Model
Ring torus volume and surface from major R and tube r.
Scope
Ring torus R > r
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Embedded snapshot: unpublished · Build schema 1.18.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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 · 111/111 tests · Production surface contract 3/3 · 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: 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=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.