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Regular Octahedron — Definition, Formula & Examples

A regular octahedron is a three-dimensional solid made of 8 identical equilateral triangles. It has 6 vertices, 12 edges, and looks like two square-based pyramids glued together at their bases.

A regular octahedron is a convex Platonic solid bounded by eight congruent equilateral triangular faces, with four faces meeting at each of its six vertices. It possesses full octahedral symmetry and is the dual polyhedron of the cube.

Key Formula

V=23 a3A=23 a2V = \frac{\sqrt{2}}{3}\,a^3 \qquad A = 2\sqrt{3}\,a^2
Where:
  • VV = Volume of the regular octahedron
  • AA = Total surface area of the regular octahedron
  • aa = Length of one edge

Worked Example

Problem: Find the surface area and volume of a regular octahedron with edge length 4 cm.
Surface Area: Substitute a = 4 into the surface area formula.
A=23 (4)2=23 (16)=323≈55.4 cm2A = 2\sqrt{3}\,(4)^2 = 2\sqrt{3}\,(16) = 32\sqrt{3} \approx 55.4 \text{ cm}^2
Volume: Substitute a = 4 into the volume formula.
V=23 (4)3=23 (64)=6423≈30.2 cm3V = \frac{\sqrt{2}}{3}\,(4)^3 = \frac{\sqrt{2}}{3}\,(64) = \frac{64\sqrt{2}}{3} \approx 30.2 \text{ cm}^3
Answer: The surface area is 323≈55.432\sqrt{3} \approx 55.4 cm² and the volume is 6423≈30.2\frac{64\sqrt{2}}{3} \approx 30.2 cm³.

Why It Matters

The octahedron appears in chemistry as the shape of molecules with octahedral geometry, such as sulfur hexafluoride (SF₆). Understanding its properties also helps in courses on solid geometry, crystallography, and when verifying Euler's formula for polyhedra.

Common Mistakes

Mistake: Confusing a regular octahedron (8 triangular faces) with a regular hexahedron, or cube (6 square faces).
Correction: Remember that "octa" means eight. A regular octahedron has 8 equilateral triangle faces, while a hexahedron (cube) has 6 square faces. They are actually duals of each other.