Surface Area of an Octahedron

What is a regular octahedron?

A regular octahedron is one of the five Platonic solids. It has 8 equilateral triangular faces, 6 vertices, and 12 equal edges. It looks like two regular square pyramids joined at their common square base.

Octahedron surface area formula

The total surface area of a regular octahedron (all 8 triangular faces combined):

$$\mathrm{SA}=2\sqrt{3}\cdot a^2$$

The apothem of each face (height of the equilateral triangle with side a):

$$h_f=\frac{a\sqrt{3}}{2}$$

where a is the edge length of the octahedron.

Applications of octahedron surface area calculations

• Crystallography and mineralogy. The octahedral shape of natural crystals is found in diamonds, fluorite, magnetite, and alum crystals. Calculating the surface area of a crystal helps analyze its dissolution rate, chemical reactivity, and optical properties.
• Chemistry and molecular geometry. The sulfur hexafluoride (SF₆) molecule has a regular octahedral shape — a central sulfur atom surrounded by six fluorine atoms. The surface area of molecular models is used to calculate the contact area in chemical reactions.
• Manufacturing and coatings. Octahedral parts (e.g., jewelry or decorative elements) require precise surface area calculation to determine the consumption of gold, silver, or protective lacquer during coating.
• Tabletop games and design. The eight-sided die (d8) used in role-playing games (like Dungeons & Dragons) is a regular octahedron. Manufacturers calculate the surface area to estimate plastic or resin consumption per unit.
• Architecture and geodesic structures. Octahedral elements are used in space frames and geodesic domes due to their high rigidity with minimal material usage. The surface area is needed to calculate the cladding or glazing of facade panels.
• 3D printing and jewelry design. In manufacturing octahedral pendants, models, or prototypes, the surface area determines the consumption of material (resin, metal, or plastic) and helps estimate the production cost.

How to use the calculator

1. Enter the edge length of the octahedron a.
2. Select the required decimal precision.
3. Click "Calculate" — the surface area and face apothem will appear instantly.

FAQ

What is the area of one face of a regular octahedron?
Each of the 8 faces is an equilateral triangle with side a. Area of one face: S_f = (√3/4)·a². Total surface area: SA = 8·S_f = 2√3·a².

How do the surface areas of an octahedron and tetrahedron compare for the same edge?
SA_tetrahedron = √3·a², SA_octahedron = 2√3·a². The octahedron's surface area is exactly twice that of a tetrahedron with the same edge — which makes sense since an octahedron has 8 faces versus 4 for a tetrahedron.

Does the surface area depend on the orientation of the octahedron?
No. The surface area is an intrinsic geometric property that depends only on the edge length a and does not change when the octahedron is rotated or reflected.