CalcKit
Octahedron Calculator
Table of Contents
Octahedron definition
An octahedron is a three-dimensional shape with eight faces. Its name comes from Greek and means "eight faces".
This tool works with the regular octahedron, whose eight faces are identical equilateral triangles. It looks like two square pyramids joined at their bases, one pointing up and one pointing down.
A regular octahedron has the following properties:
- 8 faces, all equilateral triangles
- 12 edges of equal length
- 6 vertices, with four faces meeting at each one
- The angle between two adjacent faces (the dihedral angle) is about 109.47°
- Its three diagonals, joining opposite vertices, are equal and cross at right angles in its centre
The regular octahedron is one of the five Platonic solids. It is the dual of the cube: placing a point at the centre of each face of a cube and joining them gives an octahedron, and the other way round.
Parts of an octahedron
The five parts of an octahedron mentioned in this tool are the side (a), area (A), volume (V), inradius (r) and circumradius (R).
Side (a)
The side (or edge length) of a regular octahedron is the length of any of its twelve edges.
It can be found with the following formulas:
-
If you know the inradius:
a = √6 * r -
If you know the circumradius:
a = √2 * R -
If you know the surface area:
a = √(A / (2 * √3)) -
If you know the volume:
a = ³√(3 * V / √2)
Area (A)
The area (total surface area) of a regular octahedron is the combined area of its eight equilateral triangles. Each face has an area of √3 * a² / 4, so:
A = 2 * √3 * a²
Volume (V)
The volume of a regular octahedron is the amount of space inside it. As two square pyramids, each with a base of a² and a height of a / √2:
V = √2 * a³ / 3
Inradius (r)
The inradius of an octahedron is the radius of the largest sphere that fits inside it, touching all eight faces:
r = a * √6 / 6
Circumradius (R)
The circumradius of an octahedron is the radius of the sphere that passes through all six of its vertices. It is half the distance between two opposite vertices:
R = a * √2 / 2
The Platonic solids
A Platonic solid is a convex shape whose faces are all the same regular polygon, with the same number of faces meeting at every vertex. There are only five:
- Tetrahedron: 4 triangular faces
- Cube: 6 square faces
- Octahedron: 8 triangular faces
- Dodecahedron: 12 pentagonal faces
- Icosahedron: 20 triangular faces
Octahedrons occur in nature as the crystals of diamond, fluorite and magnetite, and they are the shape of the eight-sided die (d8) used in tabletop games.
Frequently asked questions
How do you find the volume of an octahedron?
For a regular octahedron, use V = √2 * a³ / 3. For a = 3, that is √2 × 3³ / 3 ≈ 12.73 cubic units.
How do you find the surface area of an octahedron?
Multiply the square of the side by 2 * √3: A = 2 * √3 * a². For a = 3, that is 2 × √3 × 3² ≈ 31.18 square units.
How many faces, edges and vertices does an octahedron have?
8 faces, 12 edges and 6 vertices. Like every convex polyhedron, it follows Euler's formula: 6 - 12 + 8 = 2.
Is an octahedron two pyramids?
Yes. A regular octahedron is two square pyramids with equilateral side faces, joined base to base. Its volume is twice the volume of one of them.