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Icosahedron

An icosahedron is a three-dimensional shape with twenty faces which makes it a polyhedron. It is one of the few platonic solids. In this lesson, we will discuss more about the shape of Icosahedron and formulas related to it with the help of solved examples. Stay tuned to learn more!

1.	What is Icosahedron?
2.	Truncated Icosahedron
3.	Properties of Icosahedron
4.	FAQs on Icosahedron

What is Icosahedron?

The word "Icosahedron" is made of two Greek words "Icos" which means twenty and "hedra" which means seat. The icosahedron's definition is derived from the ancient Greek words Icos (eíkosi) meaning 'twenty' and hedra (hédra) meaning 'seat'. It is one of the five platonic solids with equilateral triangular faces. Icosahedron has 20 faces, 30 edges, and 12 vertices. It is a shape with the largest volume among all platonic solids for its surface area. It has the most number of faces among all platonic solids.

Icosahedron

Icosahedron's vertices	12
Icosahedron's faces	20
Icosahedron's edges	30
Icosahedron's angles	(a)Angle between edges: 60° (b)Dihedral angel: 138.19°
Icosahedron's volume formula	(5/12) × (3+√5) × a³
Icosahedron's surface area formula	(5√3 × a²)

Tips to remember

The tetrahedron, cube, octahedron, icosahedron, and dodecahedron are the only five platonic solids.
An icosahedron is the only platonic solid with 20 faces. This is the maximum number of faces a platonic solid can have.
20 faces are all equilateral triangles, so all their corner angles are 60 degrees (π/3 radians).
An icosahedron symbolizes water.

Truncated Icosahedron

The truncated icosahedron is an Archimedean solid.
Its face has two or more types of regular polygons.
It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices, and 90 edges.
It is the shape used in constructing soccer balls where white hexagons and black pentagons are joined together.
The truncated icosahedron is the base model of the Buckminsterfullerene.

Buckminsterfullerene

Area of the Truncated Icosahedron = 72.607253 a²
Volume of the Truncated Icosahedron = 55.2877308 a³

Truncated Icosahedron Shape

Properties of Icosahedron

Some of the basic properties of this platonic solid are:

Icosahedron has 20 faces, 30 edges, and 12 vertices.
The shape consists of equilateral triangle faces.
It has the greatest volume for its surface area of any platonic solid.
It has the greatest number of faces of any platonic solid.

Topics Related to Icosahedron

Important Topics

3D Shapes

Solved Examples

Example 1: Find the volume of a fair dice shaped like an icosahedron with a side of length 5 inches.
Solution: Given, Length of the side of an icosahedron= 5 inches. We know, Volume of Icosahedron = (5/12) × (3+√5) × a³. Thus, Volume of Icosahedron = (5/12) × (3+√5) × 5³= (625/12) × (3+√5) = 272.71 in³
\(\therefore\) The volume of dice shaped like an icosahedron is 272.71 in³.
Example 2: What is the ratio of the volume to the surface area of the Icosahedron for the given value of side length?
Solution: We know that,
Volume of Icosahedron, V = (5/12) × (3+√5) × a³. Surface Area of Icosahedron, A = (5√3 × a²).Thus, the ratio of the volume to the surface area of the Icosahedron is,
\( \begin{align*} \dfrac VA &= \dfrac { \dfrac5{12} \times \left(3+√5 \right) \times a^3 } { 5 \sqrt3 \times a^2 } \\ \dfrac VA &= \left ( \dfrac { 3+√5 } {\sqrt3} \right) \times a \end{align*} \)
\(\therefore\) The ratio of the volume to the surface area of the Icosahedron is approximately 0.25a.
Example 3: Find the surface area of an icosahedron whose volume is given as 139.628 in³ and the length of a side is 4 in.
Solution: We know that,
Volume of Icosahedron, V = (5/12) × (3+√5) × a³. Surface Area of Icosahedron, A = (5√3 × a²). On dividing surface area by volume, we get A/V = 4/a . Thus, A = 4V/a = (139.628 × 4)/ 4= 139.628 in²
\(\therefore\) The surface area of icosahedron is 139.628 in².
Example 4: What is the area of the truncated Icosahedron whose side length is 2 ft?
Solution: Given a = 2 ft
Using formula, Area of the truncated icosahedron = 72.607253 a²⇒ Area = 72.607253 × 2² = 290.429 ft²
\(\therefore\) The area of icosahedron is 290.429 ft².
Example 5: Find the volume of the truncated Icoahedron whose side length is 2 ft.
Solution: Given a = 2 ft
Using formula, Volume of the truncated icosahedron = 55.2877308 a³. Area = 55.2877308 × 2³ = 442.301 ft³
\(\therefore\) The volume of icosahedron is 442.301 ft³.

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Practice Questions on Icosahedron

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FAQs on Icosahedron

Who Discovered the Icosahedron?

Athenian mathematician Theatetus (c. 417–369 BC) discovered regular icosahedron along with the octahedron. These two platonic solids are known to be discovered by Theatetus, while the other three were discovered by Plato.

What Is an Icosahedron in Geometry?

It is a platonic solid constituting 20 faces, 30 edges, and 12 vertices.

How Many Edges and Vertices Does an Icosahedron have?

An Icosahedron has 30 edges and 12 vertices.

What Is the Difference between Icosahedron and Dodecahedron?

The differences between Icosahedron and Dodecahedron are:

The number of vertices in an Icosahedron is 12 while the number of vertices in a Dodecahedron is 20.
The number of faces in an Icosahedron is 20 while the number of faces in Dodecahedron is 12.
The volume/length in an Icosahedron is 2.182 while in the case of Dodecahedron volume/length is 7.663

How Many Tetrahedrons are in an Icosahedron?

There are 5 Tetrahedrons in an Icosahedron.

What Is the Difference Between Icosahedron and Icosagon?

An icosahedron is a three-dimensional shape while Icosagon is a two-dimensional shape.

How Many Vertices Does a Truncated Icosahedron Have?

A Truncated Icosahedron has 60 vertices.

What Does Icosahedron Symbolize?

Icosahedron is considered as the fifth and final platonic solid that consists of 20 triangle-shaped sides which symbolizes the element of water.

What Does Icosahedron Look Like?

The shape icosahedron is a polyhedron i.e. a 3D shape with flat surfaces. It has 20 faces or flat surfaces, 12 vertices or corners, and 30 edges. The faces of icosahedron are shaped in equilateral triangles.

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