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Math calculator

Circle Calculator

Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. 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 · 7/7 declared partitions (from-radius, from-diameter, from-circumference, from-area, arc, sector, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Circle diameter, circumference, area, arc, and sector, with an engine-linked figure of the radius (and the central angle for arc/sector).
Scope
Euclidean plane
Verification
Engine tested · Source checked · v1.1.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 4/4 property · digest 9db7bf18f0a0
Legacy regression
10/10 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 1/1 (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
6/6 golden · 3/3 CVP boundary · 7/7 invalid · 4/4 property · 4/4 cross-interface · 3/3 cross-calculator · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — plane circle
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 6/6 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.0 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.

CircumferenceC = 2πr
AreaA = πr²
Arcs = r θ
iThis is not a² + b² = c². Right triangles stay on /calc/math/pythagorean. Sector is /calc/math/sector. Arc is /calc/math/arc.

How to use

1

Choose what you know

Radius, diameter, circumference, or area. Arc and sector also need the central angle in degrees.

2

Read r, d, C, and A together

Arc mode reports arc length; sector reports sector area. The figure tracks r and, for arc/sector, the central angle.

Example calculations

Common configurations with formula and result.

ϟ

Unit circle

r = 1

C = 2π, A = π
d=2
ϟ

Semicircle arc

r = 2, θ = 180°

s = rπ
2π

Circle calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A Euclidean circle of radius r has diameter 2r, circumference 2πr, and area πr². Arc length s = rθ (θ in radians).
What it calculates
Circle diameter, circumference, area, arc, and sector, with an engine-linked figure of the radius (and the central angle for arc/sector).
Inputs
  • mode
  • r?
  • diameter?
  • circumference?
  • area?
  • theta_deg?
Outputs
  • r
  • diameter
  • circumference
  • area
Formula
C = 2πr, A = πr²
Assumptions
  • Euclidean plane
  • Not spherical geometry
  • Not Pythagorean-only
Units
  • length (consistent units)
  • degrees
Boundary conditions
  • r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
r=1 → C=2π, A=π
Validation cases

3 published on this page · 10/10 tests · Production surface contract 5/5 · View evidence

  • mode=from_radius r=1 → diameter=2
  • mode=arc r=2 theta_deg=180 → arc_length=2π
  • mode=sector r=2 theta_deg=90 → sector_area=π
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for circle radius, diameter, circumference, and area
  • CRC Standard Mathematical Tables — plane circle
    Supports: C = 2πr, A = πr², arc s = rθ, sector A = (1/2)r²θ with θ in radians
Calculation version
1.1.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator, Parallelogram Calculator . Explore all Geometry.

→ 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. › → Arc Calculator Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. 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. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. Not CAS. Runs locally. › → Hyperbola Calculator Standard hyperbola x²/a² − y²/b² = 1 (horizontal) or y²/a² − x²/b² = 1 (vertical). Same engine as Conic Section. Not CAS. Runs locally. › → Irregular Polygon Calculator Shoelace area of a simple 3–8 vertex polygon. Same engine as Plane Geometry. Not /regular-polygon. Not a triangle solver. Not CAS. Runs locally. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. › → Parallelogram Calculator Parallelogram A = ab sin θ, P = 2(a+b), diagonals from a, b, and included angle. Same engine as Plane Geometry. Not a rectangle page. Not the fluid trapezoid. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the Pythagorean calculator?

No. Right-triangle a²+b²=c² is /calc/math/pythagorean.

Where does this run?

Locally in the browser by default. REST and MCP call the same circle engine.