Regular Octahedron Calculator
Calculate the volume, surface area, and other properties of a regular octahedron using its edge length.
Regular Octahedron Volume and Surface Area Calculator
Calculate the volume and surface area of a regular octahedron with our easy-to-use calculator. Learn about octahedron formulas, applications, and explore interactive examples.
A regular octahedron is a three-dimensional shape with eight equilateral triangular faces. It is one of the five Platonic solids and can be thought of as two square pyramids joined at their bases. Its symmetrical structure makes it important in crystallography and geometric design.
Formula
Volume = (√2/3)a³
Surface Area = 2a²√3
Where:
- a:length of any edge (all edges are equal in a regular octahedron)
How to Calculate Octahedron Volume
The volume of a regular octahedron is proportional to the cube of its edge length. The factor of √2/3 comes from the geometric relationship between the edge length and the height through the center.
Understanding Surface Area
The surface area is the sum of the areas of eight equilateral triangular faces. Each face has area (a²√3)/4, where a is the edge length. Multiplying by 8 gives the total surface area of 2a²√3.
Real-World Applications
Octahedra appear in crystal structures (like diamond and fluorite), molecular geometry, and architectural design. They're also used in gaming dice (d8) and decorative objects.
Frequently Asked Questions
What makes the octahedron special among Platonic solids?
The octahedron is dual to the cube, meaning if you place vertices at the center of each face of a cube and connect them, you get an octahedron (and vice versa). It has perfect symmetry and appears frequently in crystal structures.
What are the symmetry properties of an octahedron?
A regular octahedron has 24 rotational symmetries and 24 reflective symmetries, for a total of 48 symmetries. It has six vertices, twelve edges, and eight faces, all perfectly regular.
How is an octahedron related to a cube?
An octahedron and cube are dual polyhedra. The octahedron can be inscribed in a cube so that its vertices are at the center of each face of the cube. The octahedron's edges pass through the cube's edges at their midpoints.