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.
Enter values to calculate.
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 · 8/8 declared partitions (rectangle, square, regular-polygon, annulus, parallelogram, trapezoid, irregular-polygon, invalid-domain) · Matrix
- Known limitations
- Core CVP does not include live graph, viewport, or pointer interaction.
- Model
- Annulus area, perimeter, and an engine-linked figure of outer R and inner r.
- Scope
- Annulus R > r
- Verification
- Engine tested · Source checked · v1.5.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
- Versions
- Calculation 1.5.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
- Verification revision
- 2026-09-22.geometry-cvp · 4/4 property · digest dcd9f429cc7c
- Legacy regression
- 25/25 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
- Supplemental domain review
- Not performed
- Named expert review
- Not performed
- CVP suite
- 10/10 golden · 5/5 CVP boundary · 11/11 invalid · 4/4 property · 5/5 cross-interface · 5/5 cross-calculator · 5/5 CVP contract · Manifest
- Sources
- ISO 80000-2:2019 Quantities and units — Mathematics
- Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare)
- Evidence
- 9 legacy golden · 5 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 11/11 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Enter R and r
Outer R must be greater than inner r. Both must be positive. Default R=2, r=1.
Read A and P
2-1 annulus A = 3π, P = 6π. The figure updates with R and r.
Example calculations
Common configurations with formula and result.
2-1 ring
R = 2, r = 1
Circle rejected
R = 1, r = 1
Annulus calculator specification
Version 1.5.0 · Engine tested
- Engine tested 25/25 tests · Production surface contract 5/5
- Named expert review Not performed
- Calculation version 1.5.0
- 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, perimeter, and an engine-linked figure of 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 5/5 · View evidence
- R=2 r=1 → A=3π P=6π
- R=1 r=1 → INVALID_INPUT
- Sources
- ISO 80000-2:2019 Quantities and units — MathematicsSupports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
- Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare) — §§9.1–9.3 Polar Coordinates, Length and AreaSupports: Polar coordinates and the circular wedge area r²Δθ/2, together with area = ∫r² dθ/2 and polar arc-length integration. A full turn gives disk area πr² and circumference 2πr. An annulus follows by subtracting disk areas and adding both boundary lengths; these are derived applications. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.5.0
Background
Interpretation and common distinctions.
What does this annulus calculator compute?
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.
The published formula or method is A = π(R² − r²), P = 2π(R + r). The following references support the stated mathematics or method:
- Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare): Polar coordinates and the circular wedge area r²Δθ/2, together with area = ∫r² dθ/2 and polar arc-length integration. A full turn gives disk area πr² and circumference 2πr. An annulus follows by subtracting disk areas and adding both boundary lengths; these are derived applications. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Which inputs and limits should you check?
Enter positive radii with R > r. The perimeter includes both boundary circles. Area uses squared length units, while perimeter uses the radius’s length unit.
The worked example uses R = 2, r = 1 with A = 3π. Compare it with the Example calculations and the complete result for the same inputs.
How can you verify and cite this result?
CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.plane v1.5.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.
Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.
Related tools
Other calculators in this family: Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator, Parallelogram Calculator . Explore all Geometry.
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.