What is a regular octahedron?
A regular octahedron is a three-dimensional solid made from eight congruent equilateral triangles. It has 6 vertices, 12 edges and 8 faces. It is one of the five Platonic solids and is the geometric dual of the cube: the cube has 6 faces and 8 vertices, while the octahedron has 8 faces and 6 vertices.
Think of the octahedron as two identical square-based pyramids joined base-to-base. That single idea explains much of its geometry. The square where the pyramids meet has side length a, each triangular face is equilateral, and the distance from the center to a top or bottom vertex is the circumradius.
How to use the octahedron calculator
Choose a known value
Enter a, V, S, d, R, r or ρ. The calculator can work backwards from any one of them.
Choose units
Use cm, mm, m, inches or feet. Length, area and volume units update together.
Calculate
The formulas are applied to the regular-octahedron model, then all derived quantities are refreshed.
Understand
Use the live charts and tables to see how changing a changes every measurement.
Octahedron formulas — the complete set
For a regular octahedron with edge length a, these are the core formulas used by the calculator.
What each measurement means
Reference table| Quantity | Meaning | Formula | Scales as |
|---|---|---|---|
| Edge | Any of the 12 equal edges | a | length |
| Face area | Area of one equilateral triangle | √3a²/4 | a² |
| Surface area | Total area of all 8 triangular faces | 2√3a² | a² |
| Volume | Space enclosed by the solid | √2a³/3 | a³ |
| Midradius ρ | Center to midpoint of an edge | a/2 | a |
| Inradius r | Center to a face | a√6/6 | a |
| Circumradius R | Center to a vertex | a√2/2 | a |
Why is the volume V = √2a³ / 3?
Start with the square formed by the four middle vertices. Its side is a, so its area is a². The octahedron is two square pyramids sharing this square as a base.
The distance from the center of the square to one of its vertices is the square's half-diagonal: a/√2.
The sloping edge from a pyramid's top vertex to a square vertex has length a.
Pythagoras gives the pyramid height: h² + (a/√2)² = a², hence h = a/√2.
One pyramid has volume (1/3)a²(a/√2). Double it: V = 2a³/(3√2) = √2a³/3.
Why is the surface area 2√3a²?
Every face is an equilateral triangle. The area of one equilateral triangle is √3a²/4. There are eight faces:
Inradius, midradius and circumradius: what is the difference?
These three radii all start at the same center, but they end at different geometric features. Confusing them is one of the most common mistakes in polyhedron problems.
Inradius r
Perpendicular distance from the center to any triangular face. An insphere with radius r touches all eight faces.
Midradius ρ
Distance from the center to the midpoint of any edge. The corresponding midsphere touches all 12 edges.
Circumradius R
Distance from the center to any of the six vertices. A circumsphere passes through every vertex.
Space diagonal d
Distance between opposite vertices. Because the center is the midpoint, d = 2R.
The 109.47° dihedral angle
The dihedral angle is the angle between two adjacent triangular faces. For a regular octahedron it is arccos(−1/3) ≈ 109.47°. This is not the same as the 60° angle inside each equilateral triangular face.
Worked examples
Example 1 — edge length a = 5 cm
Volume: V = (√2/3)(5³) ≈ 58.926 cm³.
Surface area: S = 2√3(5²) ≈ 86.603 cm².
Space diagonal: d = 5√2 ≈ 7.071 cm.
Radii: r ≈ 2.041 cm, ρ = 2.500 cm, R ≈ 3.536 cm.
Example 2 — volume is known: V = 1000 cm³
Rearrange V = (√2/3)a³:
Substituting 1000 cm³ gives a ≈ 12.599 cm. Once a is known, the remaining quantities follow from the standard formulas.
Reverse formulas — start from what you know
Study guide| Known quantity | How to find edge a | Then calculate |
|---|---|---|
| Volume V | a = ∛(3V/√2) | S, d, r, ρ, R |
| Surface area S | a = √(S/(2√3)) | V, d, r, ρ, R |
| Space diagonal d | a = d/√2 | V, S, r, ρ, R |
| Circumradius R | a = R√2 | V, S, d, r, ρ |
| Inradius r | a = r√6 | V, S, d, ρ, R |
| Midradius ρ | a = 2ρ | V, S, d, r, R |
Octahedron properties you should know
Geometry at a glance
Reference| Property | Regular octahedron | Why it matters |
|---|---|---|
| Faces | 8 equilateral triangles | Surface area is 8 × one triangle |
| Edges | 12 equal edges | Every edge has length a |
| Vertices | 6 | Three pairs of opposite vertices |
| Euler characteristic | 6 − 12 + 8 = 2 | Checks the topology of a convex polyhedron |
| Face angle | 60° | Each face is equilateral |
| Dihedral angle | ≈ 109.47° | How adjacent faces meet |
| Space diagonal | a√2 | Opposite vertex to opposite vertex |
| Symmetry | Highly symmetric Platonic solid | All faces, edges and vertices are equivalent |
| Dual solid | Cube | Faces and vertices exchange roles |
Common mistakes
Frequently asked questions
A regular octahedron has 8 congruent equilateral triangular faces.
It has 12 edges and 6 vertices.
For edge length a, V = √2a³/3.
S = 2√3a², because the eight equilateral faces each have area √3a²/4.
The distance between opposite vertices is d = a√2.
The inradius ends at a face, while the circumradius ends at a vertex. Their formulas are r = a√6/6 and R = a√2/2.
Yes. In the dual relationship, the cube's 6 faces correspond to the octahedron's 6 vertices, while the cube's 8 vertices correspond to the octahedron's 8 faces.
Yes. Rearrange the volume equation to a = ∛(3V/√2).
For a regular octahedron it is arccos(−1/3) ≈ 109.47°.