Octahedron

‘Octa’ means eight, and ‘hedron’ means base or seat. So, an octahedron is a 3-dimensional solid with eight flat faces.

Definition

An octahedron is a polyhedron. It has eight faces, twelve edges, and six vertices. If we join two square pyramids at their bases, we can create a regular octahedron. It belongs to the family of five platonic solids.

Octahedron-shaped diamonds, ornaments, dice, and Rubik’s cubes are some real-life examples of an octahedron.

We can create the shape of a regular octahedron with the help of a 2-D octahedron net made from paper, as shown in the link below.

Source: mechamath.com

Properties

1. Has 8 faces. Each face is an equilateral triangle. 
2. Has 12 edges.
3. Each face is an equilateral triangle.
4. Has 6 vertices. At each vertex, four edges meet.

The angles between the edges of an octahedron are 60° each, and the dihedral angle is approximately 109.47122° or 109.28°28′.

Formulas

Volume

Find the volume of a regular octahedron with an edge of 1.7 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{\sqrt{2}}{3}a^{3}}$, here a = 1.7 cm
${\therefore V=\dfrac{\sqrt{2}}{3}\times \left( 1\cdot 7\right) ^{3}}$
= 2.32 cm³

Calculate the volume of a regular octahedron whose length of the edge is 3 in.

Solution:

As we know,
Volume (V) = ${\dfrac{\sqrt{2}}{3}a^{3}}$, here a = 3 in
${ \therefore V=\dfrac{\sqrt{2}}{3}\times 3^{3} }$
= 12.73 in³

Surface Area

Find the surface area of a regular octahedron with an edge of 2.6 cm.

Solution:

As we know,
Surface Area (SA) = ${2\sqrt{3}a^{2}}$, here a = 2.6 cm
${\therefore SA.=2\sqrt{3}\times \left( 2\cdot 6\right) ^{2}}$
= 23.42 cm²

Calculate the surface area of a regular octahedron in the given figure.

Solution:

As we know,
Surface Area (SA) = ${2\sqrt{3}a^{2}}$, here a = 4 cm
${ \therefore SA=2\sqrt{3}\times 4^{2}}$
= 55.42 cm²