Insphere Radius of Hexakis Octahedron given Short Edge Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{14 \times l_e(Short)}{10 - \sqrt{2}} \]

Insphere Radius of Hexakis Octahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Insphere Radius of Hexakis Octahedron?

The Insphere Radius of Hexakis Octahedron is defined as the radius of the sphere that is contained by the Hexakis Octahedron in such a way that all the faces just touching the sphere. It represents the largest sphere that can fit inside the polyhedron while touching all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \frac{14 \times l_e(Short)}{10 - \sqrt{2}} \]

Where:

\( r_i \) — Insphere Radius of Hexakis Octahedron (m)
\( l_e(Short) \) — Short Edge of Hexakis Octahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the radius of the inscribed sphere based on the length of the shortest edge of the Hexakis Octahedron, using mathematical constants derived from the geometric properties of the shape.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing properties, volume relationships, and spatial characteristics of complex polyhedral structures like the Hexakis Octahedron.

4. Using the Calculator

Tips: Enter the length of the short edge of the Hexakis Octahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron. It has 48 faces, 72 edges, and 26 vertices.

Q2: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron and touches all faces, while the circumsphere radius is the radius of the smallest sphere that contains the polyhedron.

Q3: What are typical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and any field dealing with complex geometric structures.

Q4: Are there limitations to this formula?
A: This formula is specific to the Hexakis Octahedron geometry and assumes a perfect mathematical shape without deformations or imperfections.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit, but you can convert your measurements to meters before inputting them for accurate results.