Volume Of Hexakis Icosahedron Given Medium Edge Calculator

Formula Used:

\[ V = \frac{25}{88} \times \sqrt{6(185+82\sqrt{5})} \times \left( \frac{22 \times l_{medium}}{3(4+\sqrt{5})} \right)^3 \]

Volume of Hexakis Icosahedron:

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1. What is Volume Of Hexakis Icosahedron?

The volume of a Hexakis Icosahedron is the quantity of three dimensional space enclosed by the entire surface of this complex polyhedron. It's an important geometric measurement for understanding the spatial properties of this shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{25}{88} \times \sqrt{6(185+82\sqrt{5})} \times \left( \frac{22 \times l_{medium}}{3(4+\sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( l_{medium} \) — Medium Edge length (m)

Explanation: This formula calculates the volume based on the medium edge length of the Hexakis Icosahedron, 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, and various scientific fields. For complex polyhedra like the Hexakis Icosahedron, volume calculations help in understanding spatial properties, material requirements, and structural characteristics.

4. Using the Calculator

Tips: Enter the medium edge length in meters. The value must be positive and valid. The calculator will compute the volume based on the provided edge length.

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. It has 120 faces, 180 edges, and 62 vertices.

Q2: What are the different edge types in a Hexakis Icosahedron?
A: A Hexakis Icosahedron has three types of edges: short, medium, and long edges, each with different lengths and geometric relationships.

Q3: Can this formula be used for other edge types?
A: No, this specific formula is designed for calculating volume when the medium edge length is known. Different formulas exist for other edge types.

Q4: What are practical applications of Hexakis Icosahedron volume calculations?
A: These calculations are used in crystallography, architectural design, molecular modeling, and various fields where complex geometric shapes are studied.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the Hexakis Icosahedron, assuming exact input values and proper implementation of the formula.