Octahedron

Octahedron
Regular Octahedron

A polyhedron with eight faces. A regular octahedron has faces that are all equilateral triangles. A regular octahedron looks like it was made by gluing together the bases of two square-based pyramids.

Note: It is one of the five platonic solids.

Regular Octahedron

a = length of an edge

Volume =

Surface Area =

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Key Formula

V = \frac{\sqrt{2}}{3}\,a^3 \qquad\qquad SA = 2\sqrt{3}\,a^2

Where:

$a$ = Length of one edge of the regular octahedron
$V$ = Volume of the regular octahedron
$SA$ = Total surface area of the regular octahedron

Worked Example

Problem: Find the volume and surface area of a regular octahedron with edge length a = 6 cm.

Step 1: Write down the volume formula and substitute a = 6.

V = \frac{\sqrt{2}}{3}\,a^3 = \frac{\sqrt{2}}{3}\,(6)^3

Step 2: Compute 6 cubed.

6^3 = 216

Step 3: Multiply to get the volume.

V = \frac{\sqrt{2}}{3} \times 216 = 72\sqrt{2} \approx 101.82 \text{ cm}^3

Step 4: Now use the surface area formula with a = 6.

SA = 2\sqrt{3}\,a^2 = 2\sqrt{3}\,(6)^2 = 2\sqrt{3} \times 36 = 72\sqrt{3}

Step 5: Evaluate the surface area numerically.

SA = 72\sqrt{3} \approx 124.71 \text{ cm}^2

Answer: Volume ≈ 101.82 cm³ and Surface Area ≈ 124.71 cm² (exact values: 72√2 cm³ and 72√3 cm²).

Another Example

This example works backward from a known surface area to find the edge length, then computes the volume — the reverse direction compared to the first example.

Problem: A regular octahedron has a surface area of 200√3 cm². Find its edge length and then its volume.

Step 1: Start from the surface area formula and solve for a.

SA = 2\sqrt{3}\,a^2 \implies 200\sqrt{3} = 2\sqrt{3}\,a^2

Step 2: Divide both sides by 2√3.

a^2 = \frac{200\sqrt{3}}{2\sqrt{3}} = 100

Step 3: Take the positive square root.

a = 10 \text{ cm}

Step 4: Now substitute a = 10 into the volume formula.

V = \frac{\sqrt{2}}{3}\,(10)^3 = \frac{1000\sqrt{2}}{3} \approx 471.40 \text{ cm}^3

Answer: The edge length is 10 cm and the volume is 1000√2 / 3 ≈ 471.40 cm³.

Frequently Asked Questions

How many faces, edges, and vertices does a regular octahedron have?

A regular octahedron has 8 equilateral-triangle faces, 12 edges, and 6 vertices. You can verify this with Euler's formula for polyhedra: V − E + F = 6 − 12 + 8 = 2, which checks out.

Why does a regular octahedron look like two pyramids glued together?

If you slice a regular octahedron through its square cross-section (the widest middle), you get two congruent square-based pyramids. Each pyramid has a square base and four equilateral-triangle faces. Gluing these two bases together reconstructs the full octahedron, which is why the shape is sometimes called a "bipyramid."

What is the difference between an octahedron and a regular octahedron?

An octahedron is any polyhedron with exactly eight faces; these faces can be various shapes and sizes. A regular octahedron is the special case where all eight faces are congruent equilateral triangles and every vertex looks identical. When people say "octahedron" in geometry class, they almost always mean the regular one.

Regular Octahedron vs. Cube (Regular Hexahedron)

	Regular Octahedron	Cube (Regular Hexahedron)
Faces	8 equilateral triangles	6 squares
Edges	12	12
Vertices	6	8
Volume formula	V = (√2 / 3) a³	V = a³
Surface area formula	SA = 2√3 · a²	SA = 6a²
Dual polyhedron	Cube	Octahedron
Platonic solid?	Yes	Yes

Why It Matters

The regular octahedron appears in chemistry as the shape of many molecular structures (such as SF₆), making it essential for understanding spatial geometry. It shows up in geometry courses when studying Platonic solids, Euler's formula, and three-dimensional symmetry. Many standardized tests and math competitions include problems involving octahedron volume and surface area, so knowing the formulas saves significant time.

Common Mistakes

Mistake: Confusing the octahedron with an octagon. An octagon is a 2D polygon with 8 sides; an octahedron is a 3D polyhedron with 8 faces.

Correction: Remember: "-gon" refers to a flat polygon (2D), while "-hedron" refers to a solid polyhedron (3D). Octa- means eight in both cases, but the suffix tells you the dimension.

Mistake: Using the wrong coefficient in the volume formula, such as writing √2 / 2 instead of √2 / 3.

Correction: The correct volume formula is V = (√2 / 3) a³. One way to remember: an octahedron is two square pyramids, so you can derive the volume as 2 × (1/3 × a² × a√2 / 2) = (√2 / 3) a³.

Related Terms

Polyhedron — General category that includes octahedra
Platonic Solids — The five regular polyhedra, including the octahedron
Equilateral Triangle — Shape of each face of a regular octahedron
Right Regular Pyramid — Half of an octahedron is a square-based pyramid
Volume — Measure of space inside the octahedron
Surface Area — Total area of all eight triangular faces
Edge of a Polyhedron — A regular octahedron has 12 equal edges
Square — Cross-section shape at the octahedron's midplane