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Math calculator

Icosahedron Calculator

Regular icosahedron V = (5/12)(3+√5)a³ and A = 5√3 a² from one edge. Same engine as Solid Geometry. Not an octahedron. Not a dodecahedron. 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 · 21/21 declared partitions (sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular-prism, regular-pyramid, regular-frustum, ellipsoid, spherical-cap, spherical-zone, spherical-wedge, spherical-lune, spherical-sector, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Regular icosahedron volume, surface, face area, and vertex-to-vertex height from one edge.
Scope
Regular icosahedron
Verification
Engine tested · Specification checked · v1.18.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.18.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-21.solid-cvp-all-modes
Verification revision
2026-09-21.solid-cvp-all-modes · 4/4 property · digest d81459afdb09
Legacy regression
118/118 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 2/2 (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
26/26 golden · 27/27 CVP boundary · 32/32 invalid · 4/4 property · 8/8 cross-interface · 2/2 cross-calculator · 3/3 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
43 legacy golden · 25 legacy boundary · legacy regression suite · 26/26 oracle-backed golden · 32/32 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.18.0 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.

VolumeV = (5/12)(3+√5)a³
Total surfaceA = 5√3 a²
iSame engine as /solid-geometry. The regular octahedron is /octahedron. The regular dodecahedron is /dodecahedron. Not a second engine.

How to use

1

Enter a

Edge length must be positive. Default a = 1 is the unit regular icosahedron.

2

Read V, A, and h

Unit edge has V = (5/12)(3+√5) and A = 5√3. Edge 2 has volume (10/3)(3+√5). Height is vertex to opposite vertex, twice the circumradius.

Example calculations

Common configurations with formula and result.

ϟ

Unit edge

a = 1

V = (5/12)(3+√5)
(5/12)(3+√5)
ϟ

a = 2

a = 2

V = (10/3)(3+√5)
(10/3)(3+√5)

Icosahedron calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A regular icosahedron of edge a>0 has volume (5/12)(3+√5)a³, surface 5√3 a², face area (√3/4)a², and vertex-to-vertex height (a/2)√(10+2√5). All twenty faces are equilateral triangles. The vertex figure is a regular pentagon. This page is mode=icosahedron on math.geometry.solid. Not an octahedron. Not a dodecahedron. Not CAS.
What it calculates
Regular icosahedron volume, surface, face area, and vertex-to-vertex height from one edge.
Inputs
  • a
Outputs
  • volume
  • surface
  • height
  • face
Formula
V = (5/12)(3+√5)a³, A = 5√3 a²
Assumptions
  • Regular icosahedron
  • All faces equilateral
  • Not an octahedron
  • Not a dodecahedron
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • a ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=1 → V=(5/12)(3+√5)
Validation cases

3 published on this page · 118/118 tests · Production surface contract 5/5 · View evidence

  • a=1 → V=(5/12)(3+√5) A=5√3
  • a=2 → V=(10/3)(3+√5)
  • a=0 → VALUE_MUST_BE_POSITIVE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.18.0

Other calculators in this family: Cone Calculator, Cylinder Calculator, Dodecahedron Calculator, Ellipsoid Calculator, Frustum Calculator, Octahedron Calculator, Prism Calculator, Pyramid Calculator . Explore all Solid Geometry.

→ 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. › → 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. › → Ellipsoid Calculator Ellipsoid V = 4/3 πabc and surface from the exact elliptic-integral model, numerically evaluated. a = b = c is a sphere. Same engine as Solid Geometry. Not /sphere. Not the 2-D ellipse. Not CAS. Runs locally. › → Frustum Calculator Right circular frustum V = (1/3)πh(R²+Rr+r²) from two radii and height. Same engine as Solid Geometry. Not /cone. Not ISO 5167-5. Not CAS. Runs locally. › → Octahedron Calculator Regular octahedron V = (√2/3)a³ and A = 2√3 a² from one edge. Same engine as Solid Geometry. Not a tetrahedron. Not an icosahedron. Not CAS. Runs locally. › → Prism Calculator Right rectangular prism V = abh from three edge lengths. Same engine as Solid Geometry. Not a frustum. Not a pyramid. Not CAS. Runs locally. › → Pyramid Calculator Right rectangular pyramid V = abh/3 from three lengths. Same engine as Solid Geometry. Not a prism. Not a tetrahedron. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

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

Is this an octahedron?

No. The regular octahedron stays on /calc/math/octahedron. This page is the regular icosahedron of twenty equilateral faces.

Does this include an irregular icosahedron or other Platonic solids?

The regular dodecahedron is /calc/math/dodecahedron. Regular n-gon prism is /regular-prism. Regular n-gon pyramid is /regular-pyramid. Cube is already /prism with a=b=h. Regular tetrahedron is /tetrahedron. Regular octahedron is /octahedron.

Where does this run?

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