Octahedron Calculator

Online calculator and formulas for calculating the properties of an octahedron

The Air Element - Double Pyramid of Perfect Balance!

Octahedron Calculator

The Regular Octahedron

An octahedron is a platonic solid with 8 triangular faces, 6 vertices, and 12 edges.

Enter Known Parameters
Results
Side length a:
Volume V:
Surface S:
Diagonal d:
Platonic Solid Properties
8 triangular faces 6 vertices 12 edges Dual to cube

Octahedron Structure

Octahedron

8 equilateral triangles forming a double pyramid.

Octahedron Formulas

Volume (V)
\[V=\frac{\sqrt{2}}{3} a^3 \approx 0.471 a^3\]
Surface Area (S)
\[S= 2\sqrt{3} a^2 \approx 3.464 a^2\]
Diagonal (d)
\[d = \sqrt{2} a \approx 1.414 a\]
Dual to Cube
8 faces ↔ 8 vertices
6 vertices ↔ 6 faces

Example Calculation

Given
Edge length a = 6
Volume
\[V = \frac{\sqrt{2}}{3} \times 6^3\] \[V ≈ 0.471 \times 216 ≈ 101.8\]
Surface
\[S = 2\sqrt{3} \times 6^2\] \[S ≈ 3.464 \times 36 ≈ 124.7\]
Diagonal
\[d = \sqrt{2} \times 6\] \[d ≈ 1.414 \times 6 ≈ 8.48\]
Complete Results
Volume V ≈ 101.8 Surface S ≈ 124.7 Diagonal d ≈ 8.48

About the Octahedron

The octahedron is one of the five Platonic solids, consisting of 8 equilateral triangular faces arranged in perfect symmetry. It can be visualized as two square pyramids joined at their bases, forming a double pyramid structure.

Key properties include its duality with the cube (8 faces ↔ 8 vertices), elegant formulas involving √2 and √3, and its association with the element of air in ancient Greek philosophy. The octahedron appears naturally in crystal structures, particularly in diamonds and fluorite.