What is the volume of an octahedron?

V = (√2/3) × a³, where a is the edge length. For a regular octahedron with edge 4, that's about 30.17 cubic units.

What is the surface area of an octahedron?

SA = 2√3 × a², the combined area of its 8 equilateral triangular faces. For edge 4, that's about 55.43 square units.

What is the circumsphere radius formula?

The circumsphere radius (the sphere passing through all 6 vertices) is R = a/√2. For edge 4, that's about 2.83 units.

What units does the calculator use?

Edge length and radii use your chosen linear unit. Surface area uses that unit squared, and volume uses that unit cubed.

Octahedron Calculator — Volume & Surface Area

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Walk-through

How to Use This Calculator

3 steps

1

Enter the edge length

Type the edge length of the regular octahedron into the single input field, and pick a unit — inches, feet, centimeters, or meters. Because every edge of a regular octahedron is the same length, this one number is all the calculator needs. The result updates instantly as you type.

2

Read the volume

The Volume tab shows the volume in cubic units, plus the surface area and insphere radius as supporting stats below it. The interpretation line spells out the volume and surface area together in a single sentence.

3

Check surface area and radii

Switch to the Surface Area tab to see the 8-face breakdown (total surface area and area per face), or the Radii tab for the insphere radius (the largest sphere that fits inside) and the circumsphere radius (the sphere passing through all 6 vertices). Both update automatically whenever you change the edge length — no need to re-enter anything.

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Reference

Formula & Methodology

3 formulas

Volume

V = (√2/3) × a³

The volume of a regular octahedron depends only on its edge length a, cubed and scaled by √2/3 (≈ 0.4714). Geometrically, a regular octahedron can be split into two square pyramids joined at their square bases, and this closed-form formula is the result of summing their volumes.

Surface area

SA = 2√3 × a²

A regular octahedron has 8 identical equilateral triangular faces. Each face has area (√3/4) × a², so the total surface area is 8 × (√3/4) × a² = 2√3 × a².

Insphere and circumsphere radii

r = a/√6, R = a/√2

The insphere radius r is the radius of the largest sphere that fits entirely inside the octahedron, tangent to the center of every face. The circumsphere radius R is the radius of the smallest sphere that passes through all 6 vertices. Both follow directly from the octahedron's symmetric coordinate geometry — its 6 vertices sit at (±a/√2, 0, 0), (0, ±a/√2, 0), and (0, 0, ±a/√2) when centered at the origin.

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Glossary

Key Terms Explained

7 terms

Octahedron A three-dimensional shape with 8 triangular faces, 6 vertices, and 12 edges. A regular octahedron looks like two square pyramids joined base-to-base, and is one of the 5 Platonic solids.

Platonic solid One of exactly 5 convex polyhedra whose faces are all identical regular polygons, with the same number of faces meeting at every vertex: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Edge One of the 12 straight line segments where two faces meet. In a regular octahedron every edge is the same length, denoted a — the single input this calculator needs.

Face One of the 8 flat triangular surfaces of the octahedron. In a regular octahedron, every face is an identical equilateral triangle with area (√3/4) × a².

Volume The amount of 3D space enclosed by the octahedron, measured in cubic units. For a regular octahedron with edge a, V = (√2/3) × a³.

Insphere The largest sphere that fits entirely inside a solid, touching the center of every face. For a regular octahedron with edge a, the insphere radius is a/√6.

Circumsphere The smallest sphere that passes through every vertex of a solid. For a regular octahedron with edge a, the circumsphere radius is a/√2.

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Scenarios

Real-World Examples

3 worked examples

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Classroom geometry problem

Edge length 4

Edge length 4 in

Volume = (√2/3) × 4³ = 0.4714 × 64 ≈ 30.17 in³. Surface area = 2√3 × 4² = 3.464 × 16 ≈ 55.43 in². Insphere radius = 4/√6 ≈ 1.63 in. Circumsphere radius = 4/√2 ≈ 2.83 in. Each of the 8 faces has an area of about 6.93 in² (55.43 ÷ 8).

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Octahedral gemstone cut

Edge length 1.2 cm

Edge length 1.2 cm

Volume = (√2/3) × 1.2³ ≈ 0.4714 × 1.728 ≈ 0.81 cm³. Surface area = 2√3 × 1.2² ≈ 3.464 × 1.44 ≈ 4.99 cm². Insphere radius ≈ 1.2/2.449 ≈ 0.49 cm — useful for estimating how a small octahedral crystal (like natural diamond or fluorite) fits inside a bezel setting.

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Scaling comparison

Edge length 8 (double the classroom example)

Edge length 8 in

Doubling the edge length from 4 to 8 multiplies the volume by 2³ = 8 (from ≈30.17 to ≈241.36 in³) and the surface area by 2² = 4 (from ≈55.43 to ≈221.7 in²), while both radii only double (insphere from ≈1.63 to ≈3.27 in, circumsphere from ≈2.83 to ≈5.66 in) — a direct illustration of how volume, area, and linear measurements scale differently.

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Deep Dive

Every octahedron measurement from a single edge length

3 chapters · 3 min read

A regular octahedron is one of the 5 Platonic solids — 8 identical equilateral triangular faces, 6 vertices, and 12 equal edges, shaped like two square pyramids joined at their bases. Because every edge is the same length, one number (the edge length a) is enough to derive the volume, surface area, and both the insphere and circumsphere radii. This calculator does that instantly and explains how each value follows from the others.

How the Octahedron Calculator works

The calculator starts from the edge length a and applies closed-form formulas derived from the octahedron's geometry. Volume is V = (√2/3) × a³ — equivalent to summing the volumes of the two square pyramids that make up the shape. Surface area is SA = 2√3 × a², the sum of the 8 equilateral triangular faces, each with area (√3/4) × a². The insphere radius (largest sphere that fits inside, touching every face) is r = a/√6, and the circumsphere radius (smallest sphere through every vertex) is R = a/√2 — both follow from placing the octahedron's 6 vertices symmetrically along the x, y, and z axes.

Inputs and what they mean

The only input is the edge length, entered in inches, feet, centimeters, or meters. Because a regular octahedron has all 12 edges equal, this single measurement fully determines the shape — there's no separate base or height to enter. The unit selector is a display label only: volume results show cubed units and surface area results show squared units automatically, but the calculator does not convert between unit systems, so keep your edge-length entry in the unit you selected.

Limits and edge cases

This calculator assumes a regular octahedron — all 8 faces are congruent equilateral triangles. An irregular octahedron (with faces of different sizes, as in some crystal or engineering contexts) needs a full 3D coordinate-based calculation instead, since volume and surface area no longer reduce to a single edge-length formula. For pyramids with a square base rather than the octahedron's dual-pyramid shape, use the dedicated Pyramid Calculator; for other Platonic solids, see the Tetrahedron Calculator (4 faces) or the Cube Calculator (6 faces).

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Questions

Frequently Asked Questions

6 questions

What is the formula for the volume of an octahedron?+

V = (√2/3) × a³, where a is the edge length of a regular octahedron. For an edge length of 4, that works out to 0.4714 × 64 ≈ 30.17 cubic units.

How many faces does an octahedron have?+

A regular octahedron has 8 equilateral triangular faces, 12 edges, and 6 vertices. It looks like two square pyramids joined base-to-base.

Is an octahedron a Platonic solid?+

Yes. A regular octahedron is one of the 5 Platonic solids, alongside the tetrahedron, cube, dodecahedron, and icosahedron — all of its faces, edges, and vertex angles are identical.

What is the formula for the surface area of an octahedron?+

SA = 2√3 × a², the combined area of the octahedron's 8 identical equilateral triangular faces. For an edge length of 4, that's about 3.464 × 16 ≈ 55.43 square units — roughly 6.93 square units per face.

What is the circumsphere radius of an octahedron?+

The circumsphere radius — the radius of the smallest sphere passing through all 6 vertices — is R = a/√2 for a regular octahedron with edge length a. For edge length 4, that's about 4/1.414 ≈ 2.83 units.

What units does the Octahedron Calculator use?+

Enter the edge length in inches, feet, centimeters, or meters — pick the unit from the dropdown. Insphere and circumsphere radius results use that same linear unit, surface area results use it squared (in², ft², cm², or m²), and volume results use it cubed (in³, ft³, cm³, or m³).

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Octahedron Calculator

Edge Length

How to Use This Calculator

Enter the edge length

Read the volume

Check surface area and radii

Formula & Methodology

Key Terms Explained

Real-World Examples

Classroom geometry problem

Octahedral gemstone cut

Scaling comparison

Every octahedron measurement from a single edge length

How the Octahedron Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Octahedron Calculator

Edge Length

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How to Use This Calculator

Enter the edge length

Read the volume

Check surface area and radii

Formula & Methodology

Key Terms Explained

Real-World Examples

Every octahedron measurement from a single edge length

How the Octahedron Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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