Icosahedron
An icosahedron is a threedimensional 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.  Solved Examples on Icosahedron 
4.  Practice Questions on Icosahedron 
5.  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's vertices  12 

Icosahedron's faces  20 
Icosahedron's edges  30 
Icosahedron's angles 
(a)Angle between edges: 60° 
Icosahedron's volume formula  (5/12) × (3+√5) × a^{3} 
Icosahedron's surface area formula 
(5√3 × a^{2}) 
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.
Area of the truncated Icosahedron = 72.607253 a^{2}
The volume of the truncated icosahedron = 55.2877308 a^{3}
Topics Related to Icosahedron
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) × a^{3}. Thus, Volume of Icosahedron = (5/12) × (3+√5) × 5^{3 }= (625/12) × (3+√5) = 272.71 in^{3 }
\(\therefore\) The volume of dice shaped like an icosahedron is 272.71 in^{3}. 
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) × a^{3}. Surface Area of Icosahedron, A = (5√3 × a^{2}).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 in^{3} and the length of a side is 4 in.
Solution: We know that,
Volume of Icosahedron, V = (5/12) × (3+√5) × a^{3}. Surface Area of Icosahedron, A = (5√3 × a^{2}). On dividing surface area by volume, we get A/V = 4/a . Thus, A = 4V/a = (139.628 × 4)/ 4^{ }= 139.628 in^{2}
\(\therefore\) The surface area of icosahedron is 139.628 in^{2}. 
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 a^{2 }⇒ Area = 72.607253 × 2^{2} = 290.429 ft^{2}
\(\therefore\) The area of icosahedron is 290.429 ft^{2}. 
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 a^{3}. Area = 55.2877308 × 2^{3} = 442.301 ft^{3}
\(\therefore\) The volume of icosahedron is 442.301 ft^{3}.
Practice Questions on Icosahedron
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 threedimensional shape while Icosagon is a twodimensional shape.
How Many Vertices Does a Truncated Icosahedron Have?
A Truncated Icosahedron has 60 vertices.
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