Short Edge of Hexakis Octahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \frac{12 \times \sqrt{543 + 176 \times \sqrt{2}}}{\text{Surface to Volume Ratio} \times \sqrt{6 \times (986 + 607 \times \sqrt{2})}} \]

Short Edge of Hexakis Octahedron:

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1. What is the Short Edge of Hexakis Octahedron?

The Short Edge of Hexakis Octahedron is the length of the shortest edge of any of the congruent triangular faces of the Hexakis Octahedron. It is an important geometric parameter in the study of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \frac{12 \times \sqrt{543 + 176 \times \sqrt{2}}}{\text{Surface to Volume Ratio} \times \sqrt{6 \times (986 + 607 \times \sqrt{2})}} \]

Where:

Surface to Volume Ratio — The ratio of total surface area to total volume of the Hexakis Octahedron (m⁻¹)
\(\sqrt{}\) — Square root function

Explanation: This formula calculates the shortest edge length based on the surface to volume ratio of the Hexakis Octahedron, incorporating various mathematical constants and operations.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is crucial for understanding the geometric properties of Hexakis Octahedron, including its symmetry, surface area, volume, and other dimensional characteristics.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and valid for accurate calculation of the short edge length.

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: Why is the surface to volume ratio important?
A: The surface to volume ratio is a fundamental geometric property that influences various physical and chemical properties of polyhedra, particularly in materials science and crystallography.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and proportions of the Hexakis Octahedron, but typically ranges from very small to relatively large values.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Hexakis Octahedron shape and may not account for irregularities or imperfections in real-world objects.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived specifically for the Hexakis Octahedron and its unique geometric properties.