Volume Of Hexakis Octahedron Given Short Edge Calculator

Volume of Hexakis Octahedron Formula:

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

Volume of Hexakis Octahedron:

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1. What is the Volume of Hexakis Octahedron?

The Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron. It has 48 congruent triangular faces, 72 edges, and 26 vertices. Calculating its volume requires specific geometric formulas.

2. How Does the Calculator Work?

The calculator uses the volume formula:

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

Where:

\( V \) — Volume of Hexakis Octahedron (m³)
\( l_{short} \) — Short Edge of Hexakis Octahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula calculates the volume based on the short edge length, incorporating mathematical constants and geometric relationships specific to the Hexakis Octahedron.

3. Importance of Volume Calculation

Details: Accurate volume calculation is essential for various applications including material science, architectural design, and geometric modeling where the Hexakis Octahedron shape is utilized.

4. Using the Calculator

Tips: Enter the short edge length in meters. The value must be positive and valid. The calculator will compute the volume using the precise mathematical formula.

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. It is the dual polyhedron of the truncated cuboctahedron.

Q2: Why is the formula so complex?
A: The complexity arises from the geometric properties of the Hexakis Octahedron, which requires precise mathematical relationships between its edges and volume.

Q3: What are the units of measurement?
A: The input should be in meters, and the output volume will be in cubic meters (m³). Consistent units must be maintained for accurate results.

Q4: Can this calculator handle very small or large values?
A: Yes, as long as the input is a positive number, the calculator can handle a wide range of values, though extremely large values may require appropriate scaling.

Q5: Is this formula accurate for all Hexakis Octahedrons?
A: Yes, this formula is mathematically derived and provides accurate volume calculations for any regular Hexakis Octahedron when given the correct short edge length.