Volume Of Hexakis Icosahedron Given Short Edge Calculator

Volume of Hexakis Icosahedron Formula:

\[ V = \frac{25}{88} \times \sqrt{6 \times (185 + 82 \times \sqrt{5})} \times \left( \frac{44 \times l_{short}}{5 \times (7 - \sqrt{5})} \right)^3 \]

Volume of Hexakis Icosahedron:

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1. What is the Volume of Hexakis Icosahedron?

The Hexakis Icosahedron is a Catalan solid with 120 faces, 180 edges, and 62 vertices. The volume represents the three-dimensional space enclosed by this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{25}{88} \times \sqrt{6 \times (185 + 82 \times \sqrt{5})} \times \left( \frac{44 \times l_{short}}{5 \times (7 - \sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( l_{short} \) — Short Edge of Hexakis Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the volume based on the length of the shortest edge of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is essential in various fields including mathematics, engineering, architecture, and 3D modeling for understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the length of the short edge in meters. The value must be positive and greater than zero. The calculator will compute the corresponding volume of the Hexakis Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron, featuring 120 triangular faces.

Q2: What units should I use for the short edge?
A: The calculator uses meters as the default unit, but you can use any consistent unit as long as you maintain consistency throughout your calculations.

Q3: Can this formula be used for any polyhedron?
A: No, this specific formula is designed exclusively for calculating the volume of a Hexakis Icosahedron given its short edge length.

Q4: How accurate is the calculation?
A: The calculation is mathematically precise based on the input value, with results rounded to 6 decimal places for readability.

Q5: What if I have the long edge instead of the short edge?
A: You would need to use a different formula or conversion factor specifically designed for calculating volume from the long edge measurement.