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.
FAQs on Octahedron
1. What exactly is an octahedron and what are its key properties?
An octahedron is a three-dimensional shape, specifically a polyhedron, that is defined by eight faces, twelve edges, and six vertices. A regular octahedron is one of the five Platonic solids and is composed of eight identical equilateral triangles, with four triangles meeting at each vertex. It can be visualized as two square pyramids joined at their bases.
2. How many faces, vertices, and edges does a regular octahedron have?
A regular octahedron has a specific count for its components, which is a key part of its definition. It consists of:
8 Faces: All are equilateral triangles.
6 Vertices: These are the corner points where the edges meet.
12 Edges: These are the line segments where the faces meet.
3. What are some real-life examples of an octahedron shape?
The octahedron shape appears in various real-world contexts. Common examples include the natural crystalline structure of minerals like diamonds and fluorite. Additionally, the 8-sided dice (d8) used in many role-playing games is a perfect octahedron. Some complex molecules also exhibit an octahedral geometry.
4. What are the formulas to calculate the surface area and volume of a regular octahedron?
To calculate the surface area and volume of a regular octahedron with an edge length of 'a', you can use the following formulas:
The Total Surface Area (TSA) is calculated by multiplying the area of one equilateral triangle face by 8. The formula is: TSA = 2√3 a².
The Volume (V) is given by the formula: V = (√2 / 3) a³.
5. How is an octahedron different from a pyramid or a prism?
An octahedron, a pyramid, and a prism are all polyhedra, but they differ significantly in their structure. A pyramid has a polygonal base and triangular faces that meet at a single point (apex). A prism has two identical, parallel bases and its other faces are parallelograms. In contrast, an octahedron does not have a distinct base and top; it is a solid made of eight triangular faces and is symmetrical about its center.
6. Why is a regular octahedron considered a Platonic solid?
A regular octahedron is classified as one of the five Platonic solids because it meets all the necessary criteria. A shape must be a convex polyhedron where:
All faces are congruent regular polygons (an octahedron has 8 identical equilateral triangles).
The same number of faces meet at every vertex (in an octahedron, exactly four triangular faces meet at each of its six vertices).
7. What does the 'net' of an octahedron look like?
The net of an octahedron is a two-dimensional pattern of eight equilateral triangles that can be folded along its edges to form the 3D shape. A common arrangement for an octahedron's net looks like two rows of four triangles, where one row is inverted and offset, resembling a series of connected diamonds. When folded, these triangles come together to form the complete solid without any gaps or overlaps.
8. Can an octahedron be formed by joining two pyramids together?
Yes, this is an excellent way to conceptualise its structure. A regular octahedron can be perfectly formed by taking two identical square-based pyramids and joining them together at their square bases. The shared square base is internal to the shape and is not considered one of the octahedron's faces. This construction method highlights its symmetry and helps in understanding its properties.
