Octahedron

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Octahedron Meaning

In the case of plane geometry, an octahedron (plural: octahedra) is basically what you will wonder a shape would be formed by joining two pyramids along with their bases. The term octahedron is derived from the Greek terminology "oktaedron" which indicates "8 faced." That being said, an octahedron is a polyhedron having 8 faces, 12 edges, and 6 vertices. It is one of the five octahedron platonic solid having equilateral triangular faces. The word is commonly used to refer to the regular octahedron, a Platonic solid made up of 8 equilateral triangles, 4 of which meet at each vertex.

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Properties of Octahedron

Following are the properties of a regular octahedron:

  • An octahedron consists of 6 vertices and each vertex meets the 4 edges.

  • Octahedron sides by faces are 8 faces and 12 edges.

  • The formula to calculate the octahedron’s volume is 2/√3×a².

  • The formula to calculate the octahedron’s volume surface area is 2×√3×a².

  • The angle between octahedron edges is 60 degrees while a dihedral angle measures 109.28 degrees.

Capped Octahedron

In chemistry, the capped octahedron is a part of molecular geometry that depicts the shape of compounds where 7 atoms or groups of atoms or ligands are organized around a central atom describing the vertices of a gyroelongated triangular pyramid. This shape consists of the symmetry of C3v and is one of the three common shapes for pentacoordinate transition metal complexes, in addition to the pentagonal bipyramid and the capped trigonal prism.

Examples of the capped octahedral molecular geometry include the heptafluoromolybdate (MoF−7) and the heptafluorotungstate (WF−7) ions.


How to Find the Area of a Regular Octahedron?

A regular octahedron is composed of 8 equilateral sides.

Suppose that the length of each side of the octahedron be 'a'

Seeing that the area of an equilateral triangle is =√3/4×side²

Area of one side of the octahedron = Area of an equilateral triangle

=√3/4×a²

Therefore,

Area of the octahedron=8×√3/4×a²

=2×√3×a²

Hence, surface area (A) of the octahedron=2×√3×a²


Elongated Octahedron                      

In geometry, an elongated octahedron is referred to as a polyhedron having14 edges, 8 faces (4 triangular, 4 isosceles trapezoidal), and 8 vertices.

A similar construction is a hexadecahedron, having twenty-four edges, sixteen triangular faces, and ten vertices. Beginning with the regular octahedron, it is elongated along one axis, adding up eight new triangles. It also consists of 2 sets of 3 coplanar equilateral triangles (each creating a half-hexagon) and hence is not a Johnson solid.


Solved Examples On Octahedron

Example:

Alex has a set of two key rings that are shaped like an octahedron. He wants to know the surface area of each Keyring. Can you calculate the surface area if the length of the keyring is 0.4 in?


Solution:

The formula for the Surface area (A) of an octahedron =2×√3×a²

Given,

a = 0.4 in

Thus,

Plugging in the values for the Surface area of the keyring, we have

=2×√3× (0.4) ²

=0.5542 in²


Example:

A metal wire of length 96 ft is bent to form an octahedron. Identify the length of each of the edges of the octahedron.


Solution:

Given, the length of the wire = 96 ft.

We are already familiar that an octahedron has 12 edges.

Thus, the length of each edge of the octahedron = 96/12= 8ft

Therefore, the length of each edge of the octahedron = 8 ft


Fun Facts

  • As per the Greek philosopher, Plato, the dodecahedron signifies the universe.

  • The cube signifies the earth.

  • Likewise, all the platonic solids signify something.

  • The octahedron cube had probably known to Plato.

  • Plato knew of a solid composed of 6 squares and 8 triangles.

FAQ (Frequently Asked Questions)

Q1. What is Meant by Cube Octahedron?

Answer: The Cubo octahedral is a polyhedron consisting of eight triangular faces and six square faces. A cuboctahedron consists of 12 identical vertices, with 2 squares and 2 triangles meeting at each, and 24 identical edges, each isolating a square from a triangle. For this reason, it is a quasiregular polyhedron, i.e. an Archimedean solid which is not only vertex-transitive but also edge-transitive. It is termed a radially equilateral convex polyhedron. As such, its dual polyhedron is the rhombic dodecahedron.

Q2. What are the Different Parts of an Octahedron?

Answer: The octahedron is a 3-dimensional object and is made up of one- and 2-dimensional parts. These parts consist of special names as stated below:


Face

The octahedron consists of 8 faces, which are all in the shape of equilateral triangles. These 8 faces are where the solid attains its name. 'Octa' means 8. These faces form the surface area of the octahedron. The square which is the base of the octahedron is not part of the surface area; hence, the base isn't also a face.


Base

The base of an octahedron is in the shape of a square. If you imagine an octahedron as two congruent square pyramids that have their bottoms touching, then the base of the octahedron will be the square between the two pyramids.


Edge

When two faces touch, the segment of the line which is formed is known as an edge. An octahedron consists of 12 edges.


Vertex

When two edges bisect they form a vertex (plural being vertices). The octahedron consists of 6 vertices. Each vertex is formed when 4 edges bisect.

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