HomeCalculatorsMathGeometryRegular Polygon Calculator
Math calculator

Regular Polygon Calculator

Regular n-gon A = (n s²)/(4 tan(π/n)) for integer n≥3. Same engine as Plane Geometry. Not a triangle solver. 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 · 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
Regular n-gon area, perimeter, apothem, interior angle, and an engine-linked figure of the circumradius.
Scope
Euclidean plane
Verification
Engine tested · Specification 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 — 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
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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
9 legacy golden · 5 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 11/11 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.0 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.

AreaA = (n s²) / (4 tan(π/n))
Interior((n − 2) / n) × 180°
iSame engine as /plane-geometry. Equilateral triangle is n=3 here; the general triangle solver stays /triangle-solver. Not a second engine.

How to use

1

Enter n and s

n must be an integer ≥ 3. n = 6, s = 1 is the unit hexagon.

2

Read A, P, and the interior angle

A regular hexagon with s=1 has P=6 and interior 120°. The figure updates with n and s.

Example calculations

Common configurations with formula and result.

ϟ

Hexagon

n = 6, s = 1

A = 3√3/2
P=6, 120°
ϟ

Equilateral triangle

n = 3, s = 2

A = √3
interior 60°

Regular Polygon calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A regular n-gon with integer n≥3 and side s>0 has area (n s²)/(4 tan(π/n)), perimeter n s, interior angle ((n−2)/n)×180°, and apothem s/(2 tan(π/n)). This page is mode=regular_polygon on math.geometry.plane. Not a triangle solver. Not CAS.
What it calculates
Regular n-gon area, perimeter, apothem, interior angle, and an engine-linked figure of the circumradius.
Inputs
  • n
  • s
Outputs
  • area
  • perimeter
  • apothem
  • interior_deg
Formula
A = (n s²)/(4 tan(π/n)), P = n s
Assumptions
  • Euclidean plane
  • Integer n ≥ 3
  • Not a triangle solver
  • Not CAS
Units
  • length (consistent units)
  • area (length²)
  • degrees
Boundary conditions
  • s ≤ 0 → VALUE_MUST_BE_POSITIVE
  • n < 3 or non-integer n → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
n=6 s=1 → P=6, interior=120°
Validation cases

2 published on this page · 25/25 tests · Production surface contract 5/5 · View evidence

  • n=6 s=1 → P=6 interior=120°
  • n=3 s=2 → interior=60°
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.5.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola 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. › → 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second plane engine?

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

Is n=3 the triangle solver?

No. n=3 is the equilateral case on this engine. SSS/SAS/ASA stay on /calc/math/triangle-solver.

Where does this run?

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