Icosahedron in 3D

Twenty triangles. A remarkably balanced solid.

An icosahedron is a polyhedron with twenty faces. This model is the regular icosahedron: every face is an equilateral triangle and five faces meet at each vertex.

On this page5 sections

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Solid view

Inspect a face

Start with one edge.

Edge length a

Units

Units relabel dimensions without conversion.

faces: 20
edges: 30
vertices: 12

Volume

58.906 units³

Surface area

77.942 units²

An icosahedron net: twenty triangles, still connected.

Separate faces show the pieces. A net shows how those pieces stay joined. Here, nineteen shared edges act as hinges while the remaining edges open into the flat pattern.

Each equilateral triangle has area √3a² / 4. Twenty copies give the total surface area 5√3a².

20 faces30 edges12 vertices

20 × 3 ÷ 2 = 30 edges; 20 × 3 ÷ 5 = 12 vertices.

Twenty numbered equilateral triangles in a connected icosahedron net. — Solid outlines mark cut edges; dashed lines mark folds. Face numbers match the model.

The whole solid and its twenty faces.

Icosahedron: annotated solid and its component faces — All twenty faces are equal triangles. The separate faces keep their model colors; the diagram shows the default edge length a = 3.

Icosahedron labeled shape diagram in a vertical layout — All twenty faces are equal triangles. The separate faces keep their model colors; the diagram shows the default edge length a = 3.

Count five around a corner.

Choose a face, then turn the model to a vertex. Five triangular corners meet there. Each contributes 60°, so the angles around that vertex total 300°. The remaining angular gap lets the surface close into a solid.

One equilateral triangle√3a² / 4

Area of one face

3.897 units²

Twenty copies make the entire surface.

Icosahedron formulas and current values
Measurement	Formula	Result
Surface area	5√3a²	77.942 units²
Volume	5(3 + √5)a³ / 12	58.906 units³
Center to a vertex	a√(10 + 2√5) / 4	2.853 units
Total edge length	30a	90 units

Questions about the shape

Is every icosahedron regular?

No. Twenty faces define an icosahedron. Equal equilateral faces and the same arrangement at every vertex define the regular model shown here.

Is this a sphere?

No. Its twenty flat faces meet along sharp edges. Rotating the model makes those planes easier to see.

What happens when the edge length doubles?

Surface area becomes four times as large and volume becomes eight times as large. The face, edge and vertex counts stay fixed.

Twenty triangles,
one connected net.

The complete 3D solid and its unfolding net connect twenty triangular faces to the volume and surface-area formulas.

Regular 3D icosahedron with edge a, 20 faces, 30 edges and 12 vertices. A large connected twenty-triangle net shows how the solid folds together; exact area and volume formulas accompany it. — Featured image · Select to enlarge

20 · 30 · 12: Faces, edges and vertices.
A connected net: Dashed shared edges are the folds.
a = edge length: All twenty faces are congruent equilateral triangles.

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Definitions, formulas and example measurements remain available without JavaScript. Enable it for the interactive model and the illustrations in the single-file version. #icosahedron-lesson [data-3d], #icosahedron-lesson [data-fullscreen], #icosahedron-lesson [data-preset] { display: none; }