Short Edge Of Hexakis Octahedron Given Volume Calculator

Formula Used:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \left( \frac{28 \times V}{\sqrt{6 \times (986 + 607 \times \sqrt{2})}} \right)^{\frac{1}{3}} \]

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 this complex polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \left( \frac{28 \times V}{\sqrt{6 \times (986 + 607 \times \sqrt{2})}} \right)^{\frac{1}{3}} \]

Where:

\( V \) — Volume of Hexakis Octahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the shortest edge length based on the volume of the Hexakis Octahedron, using mathematical constants and cube root operations.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is crucial for understanding the geometric properties of Hexakis Octahedron, including surface area calculations, structural analysis, and various engineering applications involving this specific polyhedral shape.

4. Using the Calculator

Tips: Enter the volume of Hexakis Octahedron in cubic meters. The value must be positive and valid. The calculator will compute the corresponding 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 cube. It has 48 faces, 72 edges, and 26 vertices.

Q2: Why is this formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Octahedron, which requires precise mathematical relationships between volume and edge lengths.

Q3: What are typical values for the short edge?
A: The short edge length depends on the volume. For larger volumes, the short edge will be longer, following the cube root relationship in the formula.

Q4: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, though extremely large or small values may have precision limitations due to floating-point arithmetic.

Q5: Are there other ways to calculate the short edge?
A: While this volume-based method is direct, the short edge can also be calculated from other parameters if available, such as the medium or long edges of the polyhedron.