Total Surface Area of Hexakis Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{12 \times \sqrt{543 + 176\sqrt{2}}}{RA/V \times \sqrt{6 \times (986 + 607\sqrt{2})}} \right)^2 \]

Total Surface Area (TSA):

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1. What is the Hexakis Octahedron?

The Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron. It has 48 faces, 72 edges, and 26 vertices. This polyhedron is characterized by its complex geometry and symmetrical properties.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{12 \times \sqrt{543 + 176\sqrt{2}}}{RA/V \times \sqrt{6 \times (986 + 607\sqrt{2})}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( RA/V \) — Surface to Volume Ratio of Hexakis Octahedron (1/m)

Explanation: This formula calculates the total surface area based on the surface to volume ratio, incorporating complex mathematical constants derived from the geometry of the Hexakis Octahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area of a Hexakis Octahedron is important in various fields including crystallography, materials science, and geometric modeling. It helps in understanding the physical properties and behavior of materials with this specific geometric structure.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. 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 with 48 faces, 72 edges, and 26 vertices, known for its complex symmetrical properties.

Q2: What are typical surface to volume ratio values?
A: The surface to volume ratio depends on the specific dimensions of the Hexakis Octahedron. There's no fixed "normal" value as it varies with size and proportions.

Q3: What units should I use?
A: The calculator uses meters for length units, resulting in square meters for surface area and 1/m for surface to volume ratio.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric Hexakis Octahedron. Real-world applications may require adjustments for material properties and manufacturing tolerances.

Q5: Can this be used for other polyhedra?
A: No, this specific formula is designed only for the Hexakis Octahedron. Other polyhedra have different geometric properties and require different formulas.