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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 Engine tested · Specification checked · 66/66 tests · Production surface contract 3/3 · v1.11.0
Input interpretation
Enter values to calculate.
Result
—
Model
Regular icosahedron volume, surface, face area, and vertex-to-vertex height from one edge.
Scope
Regular icosahedron
Verification
Engine tested · 66/66 tests · Production surface contract 3/3 · Specification checked · v1.11.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
25 golden · 13 boundary · 28 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.11.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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 1.11.0 · 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 · 66/66 tests · Production surface contract 3/3 · 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
1.11.0

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 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.