Octahedron

Elementary School

Definition

An 8 faced polyhedron, a 3D shape with 8 faces. A regular octahedron will have 8 equal faces formed by equilateral triangles. Imagine it as two square pyramids stuck together from the bottom. It is one of the 5 platonic solids. The volume is defined as :V = \(\sqrt2 \over 3\)a³, and the surface area is defined as : \(A=2\sqrt3 a^3\)

Worked examples

\(V = \frac{\sqrt{2}}{3}a^3\)

For a regular octahedron with edge length \(a\), use this formula to find the volume.

\(A = 2\sqrt{3}a^2\)

The surface area comes from 8 equilateral triangle faces, each with area \(\frac{\sqrt{3}}{4}a^2\).

Vertices: 6, Edges: 12, Faces: 8 \(\)→\( 6 - 12 + 8 = 2\)

Euler's formula \(V - E + F = 2\) holds for this platonic solid.

Common mistakes

\(A = 2\sqrt{3}a^3\) → \(A = 2\sqrt{3}a^2\) Surface area is measured in square units, so the exponent on \(a\) must be 2, not 3.
An octahedron has 8 vertices because 'octa' means 8. → An octahedron has 8 faces and 6 vertices. The 'octa' prefix refers to faces, not vertices; a regular octahedron has 6 vertices.

Where you'll use it next

You'll use octahedra when studying Platonic solids, crystal structures in chemistry, volume and surface area formulas for polyhedra, and Euler's formula in geometry and topology.

Found in 1 StudyPug lesson

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