Octahedron Volume Calculator – Regular Polyhedron Properties
A regular octahedron is one of the five classical Platonic solids. It has 8 equilateral triangular faces, 12 edges of equal length, and 6 vertices — making it one of the most symmetric shapes in three-dimensional geometry. This calculator lets you find the volume, surface area, and all three characteristic radii of a regular octahedron from a single edge length input.
Volume Formula
For a regular octahedron with edge length a, the volume is:
V = (√2 / 3) × a³
This formula arises because a regular octahedron can be decomposed into two congruent square pyramids joined at their common square base. Each pyramid has a square base of side length a and a perpendicular height of a / √2. Applying the standard pyramid volume formula V = (1/3) × base area × height to each pyramid and summing gives the result above.
Example: if a = 5 cm, then V = (√2 / 3) × 5³ ≈ 58.926 cm³.
Surface Area Formula
The total surface area of all eight equilateral triangular faces is:
A = 2√3 × a²
Each equilateral triangular face has area (√3 / 4) × a², and there are 8 faces, so A = 8 × (√3 / 4) × a² = 2√3 × a². For a = 5 cm, the surface area is approximately 173.205 cm².
Characteristic Radii
A regular octahedron has three naturally defined spheres, each with a different relationship to the solid:
- Inradius (r = a / √6) — the radius of the inscribed sphere (insphere). This sphere sits inside the octahedron and touches the centre of every face. For
a = 5 cm,r ≈ 2.041 cm. - Midradius (ρ = a / √2) — the radius of the midsphere. This sphere passes through the midpoint of every one of the 12 edges. For
a = 5 cm,ρ ≈ 3.536 cm. - Circumradius (R = a√2 / 2) — the radius of the circumscribed sphere (circumsphere). This sphere passes through all six vertices. For
a = 5 cm,R ≈ 3.536 cm.
Note that the midradius and circumradius happen to be equal for a regular octahedron — both equal a / √2 — a consequence of its high symmetry.
Supported Units
The calculator accepts edge lengths in millimetres (mm), centimetres (cm), metres (m), inches (in), and feet (ft). The volume is expressed in the corresponding cubic unit and the surface area in the corresponding square unit. All radii share the same linear unit as the input. No cross-unit conversion is performed internally — select the unit that matches your measurement.
How to Use This Calculator
- Enter the edge length of the regular octahedron in the input field. Any positive real number is valid.
- Select the unit from the dropdown — millimetres, centimetres, metres, inches, or feet.
- Results appear instantly below, showing volume, surface area, inradius, midradius, and circumradius.
- Click the copy icon next to any result to copy that individual value, or use Copy All to copy the complete result set to your clipboard.
- Click Reset to restore the default example values.
Applications
Octahedra appear in a surprising variety of contexts. In chemistry, many crystal structures and coordination compounds adopt octahedral geometry — for example, sulfur hexafluoride (SF₆) has a regular octahedral arrangement of fluorine atoms around the central sulfur. In game design, the eight-sided die (d8) used in tabletop role-playing games is an octahedron. In architecture and art, the octahedron's clean symmetry makes it a popular motif for decorative and structural elements. Volume and surface area calculations are useful for estimating material quantities, packing problems, and structural load estimates.
Accuracy and Limitations
All calculations use JavaScript's double-precision floating-point arithmetic (IEEE 754), which provides approximately 15 significant digits of precision. Displayed results are rounded to 4 significant figures for readability; the copy-to-clipboard value provides 6 decimal places for greater precision. The calculator is valid for any positive edge length and assumes a perfect regular octahedron — all edges equal, all faces equilateral. Irregular or truncated octahedra require different formulas not covered here.