Volume of Hexakis Octahedron given Medium Edge Calculator

Volume of Hexakis Octahedron Formula:

\[ V = \frac{\sqrt{6(986+607\sqrt{2})}}{28} \times \left( \frac{14 \times le_{Medium}}{3(1+2\sqrt{2})} \right)^3 \]

Volume of Hexakis Octahedron:

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1. What is Volume of Hexakis Octahedron?

The Volume of Hexakis Octahedron is the quantity of three dimensional space enclosed by the entire surface of a Hexakis Octahedron. It represents the total capacity or space occupied by this complex polyhedral shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{\sqrt{6(986+607\sqrt{2})}}{28} \times \left( \frac{14 \times le_{Medium}}{3(1+2\sqrt{2})} \right)^3 \]

Where:

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

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

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in mathematics, engineering, architecture, and various scientific fields. For complex polyhedra like the Hexakis Octahedron, volume calculation helps in understanding spatial properties, material requirements, and structural characteristics.

4. Using the Calculator

Tips: Enter the medium edge length of the Hexakis Octahedron in meters. The value must be positive and greater than zero. 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 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 to accurately calculate its volume based on edge measurements.

Q3: What units should I use for input?
A: The calculator uses meters as the default unit for length measurements. Ensure consistent units throughout your calculations.

Q4: Can this calculator handle very small or large values?
A: Yes, the calculator can handle a wide range of values, but extremely small values may approach computational limits of floating-point precision.

Q5: Is this formula applicable to all Hexakis Octahedrons?
A: Yes, this formula is valid for all regular Hexakis Octahedrons where all medium edges are equal in length.