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Icosahedron Calculator

Find the volume of a regular icosahedron.

How it is calculated

Enter the edge length a of a regular icosahedron (all 30 edges equal). The tool computes V = (5/12) × (3 + √5) × a³, where the constant works out to about 2.1817, and the total surface area A = 5√3 × a², about 8.6603 × a², which is twenty equilateral triangles. Both results are rounded to two decimals.

The edge must be greater than zero. The answer is labelled 'units' because it carries whatever unit you used for the edge: with an edge in cm the volume is in cm³ and the area in cm². Because volume grows with the cube of the edge, doubling the edge multiplies the volume by 8 and the area by 4. The formulas apply only to the regular icosahedron, not to stretched or irregular twenty-faced shapes.

Worked example

Take an edge of a = 3. Cube it: 3³ = 27. The constant (5/12) × (3 + √5) = (5/12) × 5.2361 = 2.1817. Volume: 2.1817 × 27 = 58.91 units³. For the surface, a² = 9 and 5√3 = 8.6603, so the area is 8.6603 × 9 = 77.94 units². If the edge were in centimetres, this die-like solid holds about 58.9 cm³.

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Common mistakes

  • Using the distance across the solid (vertex to opposite vertex) instead of the edge length.
  • Mixing units, such as an edge in millimetres and then reading the result as cm³. A volume in cubic units scales with the cube of the conversion factor.
  • Applying the formula to an icosahedron with unequal edges, which is not a regular one.
  • Squaring the edge for the volume or cubing it for the area. Volume uses a³ and area uses a².
  • Confusing it with the dodecahedron (12 pentagon faces), which has different constants.

About this calculator

A regular icosahedron is a Platonic solid made of twenty equilateral triangular faces, thirty edges and twelve vertices where five faces meet at each point. Its volume follows V = (5/12)(3+√5)·a³ and its surface area is 5√3·a², both governed entirely by the edge length a because of the solid's uniform symmetry; it is the dual of the regular dodecahedron.

Structural biologists use icosahedral symmetry to describe the protein shells of many viruses, including adenovirus and herpes simplex virus, since this symmetry lets a closed shell be assembled from many copies of a few identical proteins. Architects and engineers borrow the same geometry for geodesic domes popularized by Buckminster Fuller, while game designers use the icosahedron as the classic twenty-sided die.

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Comments (1)

  • Editorial team

    Doubling the edge multiplies the surface by 4 and the volume by 8. From the unit edge result of 2.18 volume and 8.66 surface you can scale anything: an edge of 20 gives 17,453.56 cubic units and 3,464.10 square units. This also applies to a twenty-sided die. What are you building or modelling with it?

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