In geometry, a truncated octahedron is an archimedean solid. Truncating means cutting the vertices of a polyhedron. When each of the 6 vertices of a regular octahedron is cut or truncated, a truncated octahedron is formed. Each of the removed corners is a square pyramid. Thus, the final shape formed has 14 faces (8 regular hexagonal faces and 6 square faces), 36 edges and 24 vertices. Like other solids, a truncated octahedron also has volume and surface area. Let us learn more about its structure, formulas, net, along with examples in this article.
What is a Truncated Octahedron?
A truncated octahedron is a solid shape that has 6 square faces and 8 hexagonal faces. An octahedron is a polyhedron that has 8 triangular faces, 12 edges and 6 vertices. A truncated octahedron is formed by removing the 6 pyramids from the vertices of a regular octahedron. Hence, the number of surfaces is increased from 8 to 14. Also, the number of edges and vertices for the new solid formed becomes 36 and 24, respectively. In Chemistry, the structure of Faujasite is similar to that of a regular truncated octahedron.
Properties of Truncated Octahedron
- It has 14 faces (8 hexagonal and 6 square)
- It has 36 edges
- It has 24 vertices
- Each face of the truncated octahedron is a polygon
- Each face has point symmetry or 180° rotational symmetry
Formulas of Truncated Octahedron
There are two important formulas related to truncated octahedron, they are:
- Surface area
- Volume
Surface Area of Truncated Octahedron
Suppose the length of the edge of the regular truncated octahedron is ‘a’, then its surface area (S) is given by:
S = (6 + 12√3) a2 [Square units]
Volume of Truncated Octahedron
The volume of the regular truncated octahedron is given by:
V = 8√2 a3 [Cubic units]
Where a is the edge-length of regular truncated octahedron.
Net of Truncated Octahedron
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Solved Examples
Q.1: How many square faces does a truncated octahedron have?
Solution: A truncated octahedron has 6 square faces.
Q.2: A truncated octahedron has _____ vertices.
Solution: A truncated octahedron has 24 vertices.
Q.3: What is the volume of a regular truncated octahedron with an edge length equal to 3cm?
Solution: Given,
Edge-length of truncated octahedron = 3 cm
By the formula we know;
Volume of truncated octahedron = 8√2 a3
= 8√2 (3)3
= 305 .47 cu.cm.
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