What Is the Regular Octahedron Calculator?
A regular octahedron is a 3D convex polyhedron composed of eight equilateral triangular faces, meeting four at each of its six vertices.
It is the dual polyhedron of the cube and represents the octahedral crystal system common in natural diamonds, fluorite, and magnetite.
How Does the Regular Octahedron Calculator Work?
Volume is evaluated as two square pyramids joined at their bases: V = (√2 / 3) · a³.
Total surface area spans eight equilateral triangles: A = 8 · (√3 / 4) · a² = 2√3 · a².
Circumscribed sphere radius R = (a / 2) · √2 and inscribed sphere radius r = a / √6.
Regular Octahedron Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
A regular octahedron consists of 8 congruent equilateral triangular faces, 6 vertices, and 12 equal edges.
How to Use the Regular Octahedron Calculator
- Enter the edge length a.
- Examine total 3D volume, total surface area, face area, and sphere radii.
Step-by-Step Example Calculation
Edge Length a = 5
Input Values:
Understanding Your Result
An octahedron with edge length a can be inscribed perfectly inside a cube with edge length a√2.
The inradius measures the distance from the geometric center to the center of each of the eight triangular faces.
Factors That Affect the Result
- Edge Length: Scaling edge length by factor k increases surface area by k² and volume by k³.
When Should You Use This Calculator?
- Mineralogy and crystallography calculations for diamond and spinels.
- Designing geodesic polyhedral structures and tabletop gaming dice (d8).
Assumptions & Limitations
- Assumes a regular octahedron with equal edge lengths and angles.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Exact algebraic formulation using irrational constants √2 and √3.