Octahedron Calculator

Understanding the Regular Octahedron

A regular octahedron is a classic three-dimensional geometric shape (a Platonic solid) featuring 8 equilateral triangular faces, 12 edges, and 6 vertices. Since each face is congruent and all edges share the same length, its geometric properties are highly symmetric and can be determined entirely from a single parameter, such as the edge length $a$.

Regular Octahedron Formulas

If the edge length of the octahedron is $a$, we can calculate its key metrics using the following mathematical formulas:

• Volume ($V$): The total amount of space enclosed by the octahedron: $$V = \frac{\sqrt{2}}{3} a^3 \approx 0.471405 a^3$$
• Surface Area ($S$): The combined area of all 8 triangular faces: $$S = 2\sqrt{3} a^2 \approx 3.464102 a^2$$
• Circumradius ($R_u$): The radius of the sphere passing through all 6 vertices: $$R_u = \frac{\sqrt{2}}{2} a \approx 0.707107 a$$
• Inradius ($R_i$): The radius of the sphere tangent to all 8 faces: $$R_i = \frac{\sqrt{6}}{6} a \approx 0.408248 a$$
• Midradius ($R_m$): The radius of the sphere tangent to all 12 edges: $$R_m = \frac{1}{2} a = 0.5 a$$
• Height ($H$): The distance between two opposite vertices: $$H = \sqrt{2} a \approx 1.414214 a$$

How to Use the Octahedron Calculator

This calculator allows you to input any known dimension of a regular octahedron and instantly computes all other parameters.

1. Select the known parameter from the dropdown menu (e.g., Edge Length, Volume, or Surface Area).
2. Input the value of the chosen parameter in the input field.
3. The tool will automatically calculate and display the corresponding edge length, volume, surface area, and sphere radii in real time.
4. Use the detailed details field to view the exact values and copy the output.