An icosahedron is a three-dimensionalshapewith 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!

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The word "Icosahedron" is made of two Greek words "Icos" which means twenty and "hedra" which means seat. Theicosahedron'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 withequilateral triangular faces. Icosahedron has 20faces, 30edges, and 12vertices. It is a shape with the largest volume among all platonic solids for its surface area. It has the most number of faces among allplatonic solids.

Icosahedron's verticesIcosahedron's facesIcosahedron's edgesIcosahedron's anglesIcosahedron's volume formulaIcosahedron's surface area formula
 12 20 30 (a)Angle between edges:60°(b)Dihedral angel:138.19° (5/12) ×(3+√5) × a3 (5√3×a2)

Tips to remember

An icosahedron is the only platonic solid with 20faces. 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.Itsfacehas two or more types of regular polygons.It has 12regular pentagonal faces, 20 regular hexagonal faces, 60 vertices, and 90edges.The truncated icosahedron is the base model of the Buckminsterfullerene.

Area of the truncated Icosahedron= 72.607253 a2The volume of the truncated icosahedron= 55.2877308 a3

Topics Related to Icosahedron

Important Topics
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 Icosahedronis,\beginalign* \dfrac VA &= \dfrac \dfrac512 \times \left(3+√5 \right) \times a^3 5 \sqrt3 \times a^2 \\\dfracVA &= \left ( \dfrac 3+√5 \sqrt3 \right) \times a\endalign*$$\therefore$$ 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 is139.628 in2.
Example 4:What is the area of the truncated Icosahedron whose side length is 2 ft?Solution:Given a = 2 ftUsing formula,Area of the truncated icosahedron =72.607253 a2⇒Area = 72.607253 × 22=290.429ft2$$\therefore$$ The area of icosahedron is290.429ft2.
Example 5:Find the volume of the truncated Icoahedron whose side length is 2 ft.

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Solution:Given a = 2 ftUsing formula,Volume of the truncated icosahedron =55.2877308 a3. Area = 55.2877308 × 23=442.301 ft3$$\therefore$$ The volumeof icosahedron is442.301 ft3.