Octahedron Definition Formula Surface Area Volume and Solved Examples
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 in Geometry Complete Guide with Formulas and Properties
1. What is an octahedron in geometry?
An octahedron is a three-dimensional solid with 8 triangular faces, 12 edges, and 6 vertices. It is a type of polyhedron. In its regular form, all faces are congruent equilateral triangles, and all edges are equal in length. A regular octahedron is one of the five Platonic solids, meaning all its faces and angles are identical.
2. How many faces, edges, and vertices does an octahedron have?
An octahedron has 8 faces, 12 edges, and 6 vertices. These values satisfy Euler’s formula for polyhedra: V − E + F = 2.
- V (vertices) = 6
- E (edges) = 12
- F (faces) = 8
- 6 − 12 + 8 = 2 ✔
3. What is the formula for the volume of a regular octahedron?
The volume of a regular octahedron with edge length a is V = (√2 / 3)a³. To calculate:
- Step 1: Cube the edge length → a³
- Step 2: Multiply by √2
- Step 3: Divide by 3
4. What is the surface area of a regular octahedron?
The surface area of a regular octahedron is 2√3 a², where a is the edge length. Since it has 8 equilateral triangular faces:
- Area of one triangle = (√3 / 4)a²
- Total area = 8 × (√3 / 4)a²
- Simplified result = 2√3 a²
5. Why is a regular octahedron called a Platonic solid?
A regular octahedron is called a Platonic solid because all its faces are identical regular polygons and all vertices have the same arrangement. Specifically:
- Each face is an equilateral triangle
- Four triangles meet at each vertex
- All edges are equal
6. What is the difference between a cube and an octahedron?
The main difference is that a cube has 6 square faces, while an octahedron has 8 triangular faces.
- Cube: 6 faces, 12 edges, 8 vertices
- Octahedron: 8 faces, 12 edges, 6 vertices
7. How do you draw an octahedron?
To draw a regular octahedron, sketch two square-based pyramids joined at their bases. Follow these steps:
- Step 1: Draw a square (middle section).
- Step 2: Draw a point above the center and connect it to all four square corners.
- Step 3: Draw a point below the square and connect it to all four corners.
8. What are the properties of a regular octahedron?
A regular octahedron has several key geometric properties.
- 8 equilateral triangular faces
- 12 equal edges
- 6 vertices
- Four faces meet at each vertex
- Highly symmetrical solid
9. Can you give an example of calculating the volume of an octahedron?
Yes, the volume of a regular octahedron is calculated using V = (√2 / 3)a³. Example:
- Given edge length a = 6 units
- a³ = 6³ = 216
- V = (√2 / 3) × 216
- V = 72√2 cubic units
10. What are real-life examples of an octahedron?
An octahedron appears in real life in crystals, dice, and molecular structures. Common examples include:
- Eight-sided dice (D8) used in games
- Diamond crystal shapes
- Certain chemical molecular geometries
