Octahedron Calculator
Volume and surface area of a regular octahedron from its edge.
Volume
58.9256
regular octahedron (8 faces)
Surface area
86.6025
Faces / edges / vertices
8 / 12 / 6
Circumradius
3.5355
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Octahedron Calculator works
The GKCalculators Octahedron Calculator instantly computes the volume and surface area of a regular octahedron from a single edge length, making it an essential tool for students, engineers, architects, game designers, and anyone working with 3D geometry or Platonic solids.
A regular octahedron is one of the five classical Platonic solids, defined by eight equilateral triangular faces, twelve edges of identical length, and six vertices. Because every edge is equal, the entire geometry of the shape is determined by just one measurement — the edge length — which is the only input this calculator requires. Enter your edge length in your preferred unit (millimeters, centimeters, inches, etc.) and the calculator returns both the volume and the total surface area instantly.
The surface area calculation works by recognizing that a regular octahedron is composed of eight identical equilateral triangles. The area of a single equilateral triangle with side length 'a' is (√3/4)a², so the total surface area is simply eight times that value, giving 2√3 · a². This formula is exact and closed-form, meaning there is no approximation involved — only the precision limit of your input measurement. A common mistake is confusing surface area with face area; the calculator returns the total outer surface, not the area of one triangular face.
Volume is slightly less intuitive. A regular octahedron can be visualized as two square pyramids joined at their square bases. The volume of each pyramid is (1/3) × base area × height, and when you sum both, the combined volume simplifies elegantly to (√2/3) · a³. This cubic relationship means volume scales dramatically with edge length: doubling the edge length increases volume by a factor of eight, not two. This is a critical insight for anyone sizing physical models, packaging, or structural components based on a regular octahedron geometry.
The most common user error is mixing units — for example, entering an edge in centimeters and then interpreting the volume result as cubic inches. Always confirm your input unit before reading the output, since volume will be expressed in cubic units and surface area in square units of whatever length unit you entered. Another subtle pitfall is assuming these formulas apply to irregular octahedra (solids with eight faces but unequal edges or non-equilateral faces); this calculator is specifically valid only for the regular octahedron where all edges are congruent.
Formula
V = (√2 / 3) · a³, SA = 2√3 · a²
Pro tips
- Remember that volume scales with the cube of the edge length — a 10% increase in edge length results in roughly a 33% increase in volume, not 10%. Account for this when scaling physical models or containers.
- To verify your result manually, calculate the area of one equilateral triangular face as (√3/4)a² and multiply by 8; if it matches the surface area output, your input and unit selection are correct.
- When working in real-world fabrication (3D printing, metalwork, woodworking), add a small tolerance to your edge length before calculating surface area so you account for material kerf or joinery overlap.
- For quick mental estimation, note that for an edge length of 1 unit, volume ≈ 0.4714 cubic units and surface area ≈ 3.4641 square units — these reference values let you sanity-check calculator outputs at other scales by applying the a³ and a² scaling rules respectively.
- If you need the inradius (radius of the inscribed sphere) or circumradius (radius of the circumscribed sphere) for fit or clearance calculations, note that inradius = a/√6 and circumradius = a/√2, both derivable directly from your edge length.
Key terms
- Regular Octahedron
- — A Platonic solid with eight equilateral triangular faces, twelve equal edges, and six vertices, where every face, edge, and vertex is geometrically identical.
- Edge Length (a)
- — The length of any one edge of the regular octahedron, which uniquely defines the entire shape due to its perfect symmetry.
- Surface Area
- — The total area of all eight equilateral triangular faces of the octahedron, calculated as 2√3 · a².
- Volume
- — The three-dimensional space enclosed within the octahedron, calculated as (√2 / 3) · a³.
- Platonic Solid
- — One of five convex polyhedra in 3D geometry — tetrahedron, cube, octahedron, dodecahedron, and icosahedron — in which every face is an identical regular polygon and the same number of faces meet at every vertex.
- Equilateral Triangle
- — A triangle with all three sides of equal length and all interior angles equal to 60°, which forms each of the eight faces of a regular octahedron.
