‘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

Has 8 faces. Each face is an equilateral triangle.

Has 12 edges.

Each face is an equilateral triangle.

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 109.28°.

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 cm^{3}

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 in^{3}

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 cm^{2}

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 cm^{2}