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Home Dodecahedron vs icosahedron

Dodecahedron vs Icosahedron in 3D

Twelve pentagons or twenty triangles? These two regular solids share thirty edges, but exchange their numbers of faces and vertices. Explore their surfaces, unfold their nets, and reveal the geometric connection between them.

Explore This Comparison

Dodecahedron

Regular pentagonal faces

Complete 3D dodecahedron with accurate faces and edges
12faces30edges20vertices

Icosahedron

Equilateral triangular faces

Complete 3D icosahedron with accurate faces and edges
20faces30edges12vertices

Drag to rotate · Scroll or pinch to zoom · Arrow keys rotate the focused model. Turn off Link Rotations to explore one shape at a time.

Choose What Stays Equal

Equal edge length: a = 2 cm on both solids. The dodecahedron has the larger circumradius.

Numbers stay the same.

One Shape Hiding Inside the Other

Place a point at the center of every face. Connect two points whenever their faces share an edge. The dodecahedron produces an icosahedron, and the icosahedron produces a dodecahedron. These are dual polyhedra.

Faces become vertices, vertices become faces, and each original edge corresponds to one dual edge. Both solids keep the same count of thirty edges.

The orange inner solid joins actual face centers. It has its own edge length and is smaller than the surrounding solid.

Dodecahedron Icosahedron 12 faces 30 edges 20 vertices 20 faces 30 edges 12 vertices Face ↔ vertex · Edge ↔ edge

Count the Faces. Follow the Folds.

A regular dodecahedron has twelve congruent regular pentagons, with three faces meeting at each vertex. A regular icosahedron has twenty congruent equilateral triangles, with five faces meeting at each vertex. Each net below contains every face exactly once.

Shape properties and visible face nets
Property Dodecahedron Icosahedron
Faces 12 regular pentagons 20 equilateral triangles
Edges 30 equal edges 30 equal edges
Vertices 20 vertices 12 vertices
Faces meeting at a vertex 3 pentagons 5 triangles
Angle inside each face 108° 60°
Euler’s formula 20 − 30 + 12 = 2 12 − 30 + 20 = 2
A preferred base? No. Any face can be chosen to rest on. No. Any face can be chosen to rest on.
Flat face nets
Dodecahedron net 1 2 3 4 5 6 7 8 9 10 11 12
12 regular pentagons. Dashed lines are folds.
Icosahedron net 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
20 equilateral triangles. Dashed lines are folds.

Dashed lines are hinges. These are two possible non-overlapping nets, not the only arrangements. The face numbers match Net and Separate views.

Which One Has More Volume?

First choose a size comparison. For the same edge length, the dodecahedron encloses about 3.51 times the volume and has a larger outer radius. For the same circumradius, its volume is still larger, but only by about 9.82%. More faces alone does not mean more volume. Use the switch above to compare both cases.

Dodecahedron

Edge a = 2 cm

Volume61.3 cm3
Surface area82.58 cm2
VolumeV = (15 + 7√5)a³ / 4
61.3 cm3
Surface areaS = 3√(25 + 10√5)a²
82.58 cm2
One pentagon areaA = √(25 + 10√5)a² / 4
6.88 cm2
CircumradiusR = √3(1 + √5)a / 4
2.8 cm
Total edge length30a
60 cm

Icosahedron

Edge a = 2 cm

Volume17.45 cm3
Surface area34.64 cm2
VolumeV = 5(3 + √5)a³ / 12
17.45 cm3
Surface areaS = 5√3a²
34.64 cm2
One triangle areaA = √3a² / 4
1.73 cm2
CircumradiusR = √(10 + 2√5)a / 4
1.9 cm
Total edge length30a
60 cm

These formulas apply to regular convex solids. Circumradius is the distance from the center to any vertex. Values use the selected scale and are rounded to two decimal places.

Questions About These Two Solids

Why do they both have 30 edges?

Each edge belongs to two faces. The dodecahedron gives 12 × 5 ÷ 2 = 30 edges. The icosahedron gives 20 × 3 ÷ 2 = 30 edges. Counting every face edge without dividing by two counts each shared edge twice.

Which shape has 12 faces and 20 vertices?

The regular dodecahedron has 12 pentagonal faces and 20 vertices. The regular icosahedron reverses those counts: 20 triangular faces and 12 vertices.

Are both shapes Platonic solids?

Yes. In each solid, every face is the same regular polygon, and the same number of faces meet at every vertex. They are two of the five convex Platonic solids.

Is a dodecahedron a 12-sided die?

A regular dodecahedron is the usual shape of a D12 die; a regular icosahedron is the usual shape of a D20 die. A geometric model alone does not guarantee that a manufactured die is fair.

Does an icosahedron have a base?

It has no uniquely distinguished base. All 20 faces are equivalent under its symmetries. You can choose a face to rest on, but that does not change the face count or make it a prism or pyramid.

Does duality make the two solids the same size?

No. Duality describes how faces, edges, and vertices correspond. The face-center construction produces a smaller solid inside the original. Separately, the size switch compares the two main models with equal edge lengths or equal circumradii.