Octahedron
An octahedron is a shape that is formed by joining by two pyramids at its bases. Once joined the shape converts to 8 faced, 12 edges, and 6 vertices. An octahedron is most commonly known as the regular octahedron i.e. when all the faces are of the same shape and size. But in most cases, it is not necessary that all faces have to be the same to be called an octahedron, with the same size also, the shape is still called octahedron. Let us learn more about the meaning, properties, area, and volume formulas of an octahedron.
| 1. | Meaning of Octahedron |
| 2. | Properties of an Octahedron |
| 3. | Surface Area of an Octahedron |
| 4. | Volume of an Octahedron |
| 5. | FAQs on Octahedron |
Meaning of Octahedron
The word octahedron is derived from the Greek word 'Oktaedron' which means 8 faced. An octahedron is a polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. It is one of the five platonic solids with faces that are shaped like an equilateral triangle.
Properties of an Octahedron
Mentioned below are a few properties of an octahedron:
- An octahedron has 6 vertices and at each vertex 4 edges meet.
- An octahedron has 8 faces shaped like an equilateral triangle, in the case of a regular octahedron.
- An octahedron has 12 edges.
- In a regular octahedron, the angles between the edges are measured at 60° but the dihedral angle is measured at 109.28°.
- The formula to calculate the surface area of an octahedron is 2×√3×a2.
- The formula to calculate the volume of an octahedron is √2/3 × a3.
Surface Area of an Octahedron
A regular octahedron is made up of 8 equilateral sides. The area of an octahedron is 2 multiplied by the square of the length of an edge multiplied by the square root of three. Let us find out the formula for calculating the surface area of an octahedron. Let the length of each side of the octahedron be 'a'. Since 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
Hence, Area of the octahedron = 8 × (√3/4) × a2 = 2 ×√3 × a2
Therefore, Surface Area (A) = 2 ×√3 × a2
Volume of an Octahedron
An octahedron can be divided into two pyramids which means the volume of an octahedron is twice the time of the volume of a pyramid. To find the volume of an octahedron, we can find the volume of one pyramid and then calculate it for two pyramids or an octahedron.
The volume of the pyramid = (Base area × Height)/3
The base of the pyramid is a square, hence base area = a2
Height of the pyramid (H) = \(\sqrt{a^2 - (\frac {a}{\sqrt2})^2}\)
The volume of the pyramid = \(\dfrac{a^2 \times \sqrt{a^2 - (\frac {a}{\sqrt2})^2}}{3} = \dfrac {a^3}{3 \sqrt{2}}\)
Hence, volume of the octahedron = 2 × volume of the pyramid = 2 × (a3)/3√2 =√2/3 × a3
Volume (V) = √2/3 × a3
Related Articles on Octahedron
Listed below are a few interesting topics that are related to the concept of an octahedron. Let's have a look:
FAQs on Octahedron
Is Octahedron a Prism?
No, an octahedron is not a prism. An octahedral prism is made of two octahedra that are connected to each other via 8 triangular prisms.
Is Octahedron a Pyramid?
Octahedron is a 3-dimensional geometric shape that is developed by joining the bases of 2 pyramids making it into a shape with 8 faced, 12 edges, and 6 vertices. The triangular pyramids can be considered as the faces of the octahedron.
How many Vertices does an Octahedron have?
An octahedron has 6 vertices out of which four vertices are there on the square base, 1 vertex on the top, and 1 at the bottom.
Does an Octahedron have Hexagonal Faces?
No, an octahedron does not have hexagonal faces but rather has triangular faces. A regular octahedron has equilateral triangular faces.
What is a Regular Octahedron?
A regular octahedron has all equilateral triangular faces of equal length. It is a rectified version of a tetrahedron and is considered the dual polyhedron of a cube. In a regular octahedron, all faces are the same size and shape. It is formed by joining 2 equally sized pyramids at their base.
What are the Different Parts of an Octahedron?
An octahedron is made of different parts such as:
- Face: An octahedron consists of 8 faces that are shaped like a triangle
- Base: An octahedron has a square-shaped base
- Edge: An octahedron consists of 12 edges
- Vertex: An octahedron consists of 6 vertices and each vertex is formed when 4 edges intersect.