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Math calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 25/25 tests · Production surface contract 3/3 · v1.5.0
Input interpretation
Enter values to calculate.
Result
—
Model
Annulus area and perimeter from outer R and inner r.
Scope
Annulus R > r
Verification
Engine tested · 25/25 tests · Production surface contract 3/3 · Specification checked · v1.5.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
9 golden · 5 boundary · 11 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.5.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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 = π(R² − r²)
PerimeterP = 2π(R + r)
iSame engine as /plane-geometry. Default R=2, r=1 gives A=3π. Not a second engine.

How to use

1

Enter R and r

Outer R must be greater than inner r. Both must be positive. Default R=2, r=1.

2

Read A and P

2-1 annulus A = 3π, P = 6π.

Example calculations

Common configurations with formula and result.

ϟ

2-1 ring

R = 2, r = 1

A = 3π
3π
ϟ

Circle rejected

R = 1, r = 1

INVALID_INPUT
INVALID_INPUT

Annulus calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
An annulus of outer radius R>r and inner radius r>0 has area π(R²−r²) and perimeter 2π(R+r). R=r is a circle, not an annulus. This page is mode=annulus on math.geometry.plane. Not /circle. Not /rectangle. Not CAS.
What it calculates
Annulus area and perimeter from outer R and inner r.
Inputs
  • R
  • r
Outputs
  • area
  • perimeter
  • outer_circumference
  • inner_circumference
Formula
A = π(R² − r²), P = 2π(R + r)
Assumptions
  • Annulus R > r
  • R = r is a circle, not an annulus
  • Not /circle
  • Not /rectangle
  • Not CAS
Units
  • length (consistent units)
  • area (length²)
Boundary conditions
  • R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • R ≤ r → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
R=2 r=1 → A=3π
Validation cases

2 published on this page · 25/25 tests · Production surface contract 3/3 · View evidence

  • R=2 r=1 → A=3π P=6π
  • R=1 r=1 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.5.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dodecahedron 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. › → 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. › → 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. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second plane engine?

No. It is mode=annulus on math.geometry.plane, the same engine as /calc/math/plane-geometry.

Is this the circle page?

No. Diameter, circumference, arc, and sector stay on /calc/math/circle. R=r is invalid here.

Where does this run?

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