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Dodecahedron

Dodecahedron
Regular Dodecahedron

A polyhedron with 12 faces. A regular dodecahedron has faces that are all regular pentagons.

Note: It is one of the five platonic solids.

Regular Dodecahedron

a = length of an edge

Volume = Surface area formula for a regular dodecahedron: (15 + 7√5) / 4 × a³, where a is edge length.

Surface Area = Surface area formula: 3a² times the square root of (25 + 10√5), where a is edge length.
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Key Formula

V=15+754 a3SA=325+105  a2V = \frac{15 + 7\sqrt{5}}{4}\,a^3 \qquad\qquad SA = 3\sqrt{25 + 10\sqrt{5}}\;a^2
Where:
  • aa = Length of one edge of the regular dodecahedron
  • VV = Volume of the regular dodecahedron
  • SASA = Total surface area of the regular dodecahedron

Worked Example

Problem: Find the surface area of a regular dodecahedron with edge length a = 2 cm.
Step 1: Write down the surface area formula for a regular dodecahedron.
SA=325+105  a2SA = 3\sqrt{25 + 10\sqrt{5}}\;a^2
Step 2: Substitute a = 2 into the formula.
SA=325+105  (2)2=325+105  (4)SA = 3\sqrt{25 + 10\sqrt{5}}\;(2)^2 = 3\sqrt{25 + 10\sqrt{5}}\;(4)
Step 3: Evaluate the expression under the square root. Since √5 ≈ 2.2361, we get 10√5 ≈ 22.361, so 25 + 22.361 = 47.361.
47.361≈6.882\sqrt{47.361} \approx 6.882
Step 4: Multiply to find the surface area.
SA=3×6.882×4≈82.58 cm2SA = 3 \times 6.882 \times 4 \approx 82.58 \text{ cm}^2
Answer: The surface area is approximately 82.58 cm².

Another Example

This example uses the volume formula instead of the surface area formula, and works with a different edge length to show both key dodecahedron calculations.

Problem: Find the volume of a regular dodecahedron with edge length a = 3 cm.
Step 1: Write down the volume formula for a regular dodecahedron.
V=15+754 a3V = \frac{15 + 7\sqrt{5}}{4}\,a^3
Step 2: Compute the numerator constant. Since √5 ≈ 2.2361, we get 7√5 ≈ 15.6525, so 15 + 15.6525 = 30.6525.
15+754≈30.65254≈7.6631\frac{15 + 7\sqrt{5}}{4} \approx \frac{30.6525}{4} \approx 7.6631
Step 3: Calculate a³ for a = 3.
a3=33=27a^3 = 3^3 = 27
Step 4: Multiply to find the volume.
V≈7.6631×27≈206.90 cm3V \approx 7.6631 \times 27 \approx 206.90 \text{ cm}^3
Answer: The volume is approximately 206.90 cm³.

Frequently Asked Questions

How many faces, edges, and vertices does a dodecahedron have?
A regular dodecahedron has 12 pentagonal faces, 30 edges, and 20 vertices. You can verify this with Euler's formula for polyhedra: V − E + F = 2, which gives 20 − 30 + 12 = 2.
What is the difference between a dodecahedron and an icosahedron?
A regular dodecahedron has 12 pentagonal faces, 30 edges, and 20 vertices. A regular icosahedron has 20 triangular faces, 30 edges, and 12 vertices. They are duals of each other, meaning the number of faces of one equals the number of vertices of the other, and they share the same number of edges.
What shape are the faces of a regular dodecahedron?
Every face of a regular dodecahedron is a regular pentagon — a five-sided polygon where all sides are equal and all interior angles are 108°. There are 12 such identical pentagonal faces in total.

Dodecahedron vs. Icosahedron

DodecahedronIcosahedron
Number of faces12 pentagons20 triangles
Number of edges3030
Number of vertices2012
Face shapeRegular pentagonEquilateral triangle
Volume formula15+754a3\frac{15+7\sqrt{5}}{4}a^35(3+5)12a3\frac{5(3+\sqrt{5})}{12}a^3
Dual solidIcosahedronDodecahedron

Why It Matters

The dodecahedron appears in geometry courses when studying Platonic solids and Euler's formula for polyhedra. It also shows up in real life: the classic soccer ball pattern includes pentagonal patches based on dodecahedral geometry, and 12-sided dice used in tabletop games are regular dodecahedra. Understanding its formulas gives you practice working with irrational numbers like √5, a skill needed throughout higher math.

Common Mistakes

Mistake: Confusing the dodecahedron (12 faces) with the icosahedron (20 faces).
Correction: Remember the prefix: 'dodeca-' comes from the Greek word for twelve, so a dodecahedron always has 12 faces. An icosahedron has 20 faces ('icosa-' means twenty).
Mistake: Using the wrong exponent in the formulas — applying a² in the volume formula or a³ in the surface area formula.
Correction: Surface area is always measured in square units, so the formula uses a². Volume is measured in cubic units, so the formula uses a³. Match the exponent to the type of measurement.

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