Octahedron Calculator
Solve octahedron problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Octahedron Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: V = (sqrt(2)/3) x a^3 | SA = 2 x sqrt(3) x a^2
Worked example — Volume: 471.405 cm^3 | Surface Area: 346.410 cm^2 | Circumradius: 7.071 cm
Formula
V = (sqrt(2)/3) x a^3 | SA = 2 x sqrt(3) x a^2
Where a is the edge length of the regular octahedron. V gives the volume enclosed, and SA gives the total surface area of all 8 equilateral triangular faces.
Worked Examples
Example 1: Regular Octahedron with Edge 10 cm
Problem:Calculate the volume, surface area, and circumradius of a regular octahedron with edge length 10 cm.
Solution:Volume = (sqrt(2)/3) x 10^3 = (1.41421/3) x 1000 = 471.405 cm^3 Surface Area = 2 x sqrt(3) x 10^2 = 2 x 1.73205 x 100 = 346.410 cm^2 Circumradius = 10 x sqrt(2)/2 = 10 x 0.70711 = 7.071 cm
Result:Volume: 471.405 cm^3 | Surface Area: 346.410 cm^2 | Circumradius: 7.071 cm
Example 2: Crystal Structure Analysis
Problem:An octahedral crystal has an edge length of 3 mm. Find its inradius and volume.
Solution:Inradius = 3 x sqrt(6)/6 = 3 x 2.44949/6 = 3 x 0.40825 = 1.2247 mm Volume = (sqrt(2)/3) x 3^3 = (1.41421/3) x 27 = 0.47140 x 27 = 12.728 mm^3
Result:Inradius: 1.225 mm | Volume: 12.728 mm^3
Frequently Asked Questions
What is a regular octahedron and what are its properties?
A regular octahedron is one of the five Platonic solids, consisting of eight equilateral triangular faces, twelve edges, and six vertices. Each vertex has four edges meeting at it, giving a vertex figure of a square. The octahedron can be thought of as two square pyramids joined at their bases. It is the dual polyhedron of the cube, meaning that if you place a point at the center of each face of a cube and connect them, you get an octahedron. The regular octahedron has the highest symmetry among all polyhedra with eight faces, possessing 48 symmetry operations in its symmetry group.
How do you calculate the volume of a regular octahedron?
The volume of a regular octahedron with edge length a is calculated using the formula V = (sqrt(2)/3) times a cubed. This can be derived by splitting the octahedron into two square pyramids and calculating their combined volume. Each pyramid has a square base with side length a and a height of a times sqrt(2) divided by 2. The volume of each pyramid is (1/3) times base area times height, which gives (1/3) times a squared times a times sqrt(2)/2, equaling a cubed times sqrt(2) divided by 6. Doubling this for both pyramids gives the total formula. For an edge length of 5, the volume is approximately 58.926 cubic units.
What is the surface area formula for a regular octahedron?
The surface area of a regular octahedron equals 2 times sqrt(3) times a squared, where a is the edge length. Since the octahedron has 8 equilateral triangular faces, you can also compute it as 8 times the area of a single equilateral triangle, which is 8 times (sqrt(3)/4) times a squared, simplifying to 2 times sqrt(3) times a squared. For an edge length of 5 units, the surface area equals approximately 86.603 square units. This formula is exact and applies only to regular octahedra where all faces are congruent equilateral triangles and all edges have equal length.
What is the difference between the inradius, midradius, and circumradius?
These three radii describe spheres that relate to the octahedron in different ways. The inradius (insphere radius) is the radius of the largest sphere that fits entirely inside the octahedron, tangent to every face. For edge length a, it equals a times sqrt(6) divided by 6. The midradius (midsphere radius) is the radius of the sphere tangent to every edge at its midpoint, equal to a divided by 2. The circumradius (circumsphere radius) is the radius of the smallest sphere that completely contains the octahedron, passing through all vertices, equal to a times sqrt(2) divided by 2. The ratio of circumradius to inradius is always sqrt(3).
How is the octahedron related to the cube as a dual polyhedron?
The octahedron and cube are dual polyhedra, meaning each can be derived from the other by swapping faces and vertices. The cube has 6 faces, 8 vertices, and 12 edges, while the octahedron has 8 faces, 6 vertices, and 12 edges. To construct an octahedron from a cube, place a point at the center of each of the six faces of the cube and connect adjacent centers. Conversely, placing points at the centers of the eight faces of an octahedron and connecting them creates a cube. Both share the same number of edges (12) and the same symmetry group. This duality is a fundamental concept in polyhedral geometry.
What is the dihedral angle of a regular octahedron?
The dihedral angle of a regular octahedron is approximately 109.4712 degrees, which equals the arccos of negative one-third. This is the angle between any two adjacent triangular faces measured along their shared edge. Interestingly, this is the same as the tetrahedral angle, the angle between bonds in a tetrahedral molecular geometry such as methane. The dihedral angle can be calculated using the dot product of the normal vectors to two adjacent faces. This angle is important in crystallography and molecular chemistry, where octahedral geometry describes the arrangement of six atoms or groups around a central atom.
Where do octahedra appear in nature and science?
Octahedral geometry is remarkably common in nature and scientific applications. In chemistry, many molecules adopt octahedral coordination geometry, where six ligands arrange symmetrically around a central atom. Sulfur hexafluoride (SF6) is a classic example. Many crystals, including diamond, fluorite, and certain metal oxides, contain octahedral structural units. In crystallography, the octahedron is one of the most common crystal habits. Alum crystals naturally grow as octahedra. In gaming, the octahedron is the shape of the 8-sided die (d8) used in tabletop role-playing games. Architecturally, octahedral structures provide exceptional strength-to-weight ratios.
Can an octahedron tessellate three-dimensional space?
A regular octahedron alone cannot tessellate (fill) three-dimensional space without gaps. However, octahedra can tessellate space when combined with tetrahedra in a ratio of 1 octahedron to 2 tetrahedra. This arrangement is called the octet truss or tetrahedral-octahedral honeycomb and was famously used by Alexander Graham Bell in his kite designs and by Buckminster Fuller in structural engineering. The resulting space-filling pattern is also found in the face-centered cubic (FCC) crystal structure of metals like gold, silver, copper, and aluminum. Each octahedron in this tessellation is surrounded by 8 tetrahedra sharing its faces.
What are the symmetry properties of a regular octahedron?
The regular octahedron has octahedral symmetry, designated Oh in Schoenflies notation, which includes 48 symmetry operations. These consist of 24 rotational symmetries and 24 improper rotations (rotation plus reflection). The rotation group alone includes 3 axes of fourfold symmetry (through opposite vertices), 4 axes of threefold symmetry (through opposite face centers), and 6 axes of twofold symmetry (through opposite edge midpoints). The octahedron also has 9 mirror planes. This rich symmetry makes the octahedron particularly important in crystallography, where octahedral symmetry describes one of the major crystal systems used to classify mineral structures.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.
🧮Conic Sections Calculator
Calculate conic sections with inputs, formulas, and instant results.