Insphere Radius of Hexakis Octahedron Given Total Surface Area Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \sqrt{\frac{7 \times TSA}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Insphere Radius (rᵢ):

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1. What is the Insphere Radius of Hexakis Octahedron?

The Insphere Radius of a 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 touch the sphere. It represents the largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{402 + 195\sqrt{2}}{194}}}{2} \times \sqrt{\frac{7 \times TSA}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the insphere radius based on the total surface area of the Hexakis Octahedron, using specific mathematical constants derived from the geometry of this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the internal packing properties of polyhedral structures and determining the maximum size of objects that can fit inside such shapes.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero for accurate calculation.

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 cube. It has 48 faces, 72 edges, and 26 vertices.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Octahedron, which requires precise mathematical relationships between its various dimensions.

Q3: What units should I use for input?
A: Use consistent units (preferably meters for length and square meters for area) to ensure accurate results.

Q4: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this calculation applicable to other polyhedra?
A: No, this specific formula is only valid for the Hexakis Octahedron. Other polyhedra have different formulas for calculating their insphere radii.