Icosahedron

Icosahedron
Regular Icosahedron

A polyhedron with 20 faces. A regular icosahedron has faces that are all equilateral triangles.

Note: It is one of the five platonic solids.

Regular Icosahedron

a = length of an edge

Volume =

Surface Area =

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Key Formula

V = \frac{5}{12}(3 + \sqrt{5})\,a^3 \qquad S = 5\sqrt{3}\,a^2

Where:

$a$ = Length of one edge of the regular icosahedron
$V$ = Volume of the regular icosahedron
$S$ = Total surface area of the regular icosahedron

Worked Example

Problem: Find the surface area and volume of a regular icosahedron with edge length a = 4 cm.

Step 1: Write down the surface area formula and substitute a = 4.

S = 5\sqrt{3}\,a^2 = 5\sqrt{3}\,(4)^2 = 5\sqrt{3}\,(16)

Step 2: Compute the surface area.

S = 80\sqrt{3} \approx 80 \times 1.7321 \approx 138.56 \text{ cm}^2

Step 3: Write down the volume formula and substitute a = 4.

V = \frac{5}{12}(3 + \sqrt{5})\,a^3 = \frac{5}{12}(3 + \sqrt{5})\,(4)^3

Step 4: Evaluate the constant factor. Note that √5 ≈ 2.2361, so 3 + √5 ≈ 5.2361.

V = \frac{5}{12}(5.2361)(64) = \frac{5 \times 335.11}{12}

Step 5: Finish the calculation.

V = \frac{1675.57}{12} \approx 139.63 \text{ cm}^3

Answer: The surface area is approximately 138.56 cm² and the volume is approximately 139.63 cm³.

Another Example

This example works backwards from a known surface area to find the edge length, showing how to rearrange the surface area formula.

Problem: A regular icosahedron has a surface area of 43.30 cm². Find its edge length.

Step 1: Start with the surface area formula and solve for a².

S = 5\sqrt{3}\,a^2 \quad \Longrightarrow \quad a^2 = \frac{S}{5\sqrt{3}}

Step 2: Substitute S = 43.30 and compute.

a^2 = \frac{43.30}{5 \times 1.7321} = \frac{43.30}{8.6603} \approx 5.000

Step 3: Take the square root to find the edge length.

a = \sqrt{5.000} \approx 2.236 \text{ cm}

Answer: The edge length is approximately 2.24 cm (which is √5 cm).

Frequently Asked Questions

How many faces, edges, and vertices does a regular icosahedron have?

A regular icosahedron has 20 triangular faces, 30 edges, and 12 vertices. You can verify this with Euler's formula for polyhedra: V − E + F = 12 − 30 + 20 = 2, which checks out.

What is the difference between an icosahedron and a dodecahedron?

An icosahedron has 20 equilateral triangle faces, 30 edges, and 12 vertices. A dodecahedron has 12 regular pentagon faces, 30 edges, and 20 vertices. They are actually duals of each other—the icosahedron's vertices correspond to the dodecahedron's faces, and vice versa. Both are Platonic solids.

Where do you see icosahedra in real life?

The most common example is a 20-sided die (d20) used in tabletop games. Many viruses, such as the adenovirus, have icosahedral shapes. Geodesic domes, like those designed by Buckminster Fuller, are also based on the icosahedron's geometry.

Icosahedron vs. Dodecahedron

	Icosahedron	Dodecahedron
Faces	20 equilateral triangles	12 regular pentagons
Edges	30	30
Vertices	12	20
Volume formula	V = (5/12)(3 + √5) a³	V = (15 + 7√5)/4 · a³
Surface area formula	S = 5√3 · a²	S = 3√(25 + 10√5) · a²
Dual solid	Dodecahedron	Icosahedron

Why It Matters

The icosahedron appears in geometry courses when studying Platonic solids, Euler's formula (V − E + F = 2), and three-dimensional symmetry. It shows up in science too: many virus capsids have icosahedral symmetry, and the carbon molecule C₆₀ (buckyball) is based on a truncated icosahedron—the same shape as a soccer ball. Understanding its properties builds your ability to work with surface area and volume formulas for non-standard solids.

Common Mistakes

Mistake: Confusing the icosahedron (20 triangular faces) with the dodecahedron (12 pentagonal faces).

Correction: Remember: 'icosa-' comes from the Greek word for 20, and 'dodeca-' means 12. The icosahedron has more faces but fewer vertices than the dodecahedron.

Mistake: Forgetting the (3 + √5) factor in the volume formula or using 5√3 · a³ for volume instead of surface area.

Correction: The surface area formula is S = 5√3 · a² (involves a²), while the volume formula is V = (5/12)(3 + √5) · a³ (involves a³ and the golden-ratio-related term 3 + √5). Keep the two formulas distinct.

Related Terms

Polyhedron — General class of 3D solids with flat faces
Platonic Solids — The five regular convex polyhedra, including the icosahedron
Equilateral Triangle — Shape of each face of a regular icosahedron
Face of a Polyhedron — A regular icosahedron has 20 triangular faces
Edge of a Polyhedron — A regular icosahedron has 30 edges
Volume — Measure of space inside the icosahedron
Surface Area — Total area of all 20 triangular faces