What is a regular icosahedron?

A regular icosahedron is one of the five Platonic solids. It has 20 congruent equilateral triangular faces, 30 edges, and 12 vertices. Every face, edge, and vertex is identical, making it highly symmetrical.

What is the formula for the volume of a regular icosahedron?

The exact formula is V = (5/12)(3 + √5) × a³, where a is the edge length. The constant (5/12)(3 + √5) ≈ 2.18169499 is derived from the geometry of the 20 equilateral triangular faces.

Does this work for irregular icosahedra?

No. This calculator is for regular icosahedra only, where all 20 faces are congruent equilateral triangles and all edges are equal in length. Irregular icosahedra require different methods.

What units does this calculator support?

You can enter the edge length in millimeters (mm), centimeters (cm), meters (m), inches (in), or feet (ft). The volume result is shown in the corresponding cubic unit plus common conversions.

How precise are the results?

The calculator uses the full JavaScript double-precision floating-point constant for √5, giving results accurate to at least 10 significant figures. The displayed value is rounded to 6 significant figures for readability.

Volume of a Icosahedron Calculator

Name: Volume of a Icosahedron Calculator
Availability: InStock
Author: Nham Vu

Enter the edge length of a regular icosahedron and instantly get its exact volume.

Icosahedron Dimensions

Edge Length (a)

Formula

V = (5/12)(3 + √5) × a³

Constant ≈ 2.18169499062

Results

Enter an edge length and click Calculate

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Summary

Enter the edge length of a regular icosahedron and instantly get its exact volume.

How it works

Enter the edge length of the icosahedron.
Select the unit of measurement (mm, cm, m, in, ft).
The calculator applies the formula V = (5/12)(3 + √5) × a³.
The volume is displayed in cubic units matching your input.
Additional conversions to other volume units are shown instantly.

Use cases

Calculate the volume of a 3D-printed icosahedral model.
Solve geometry homework or exam problems involving Platonic solids.
Determine the material needed to fill an icosahedral container.
Verify manual calculations with the exact formula.
Compare icosahedron volumes at different scales.
Use in engineering design involving polyhedral shapes.
Educational exploration of regular polyhedra properties.
Game design requiring precise solid geometry measurements.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu