Midsphere Radius of Hexakis Icosahedron given Volume Calculator

Formula Used:

\[ r_m = \frac{1}{8} \times \left( \frac{88}{25} \times \frac{V}{\sqrt{6 \times (185 + 82 \times \sqrt{5})}} \right)^{1/3} \times (5 + 3 \times \sqrt{5}) \]

Midsphere Radius of Hexakis Icosahedron:

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1. What is the Midsphere Radius of Hexakis Icosahedron?

The Midsphere Radius of Hexakis Icosahedron is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron become a tangent line on that sphere. It represents the distance from the center of the polyhedron to the point where the sphere touches all edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1}{8} \times \left( \frac{88}{25} \times \frac{V}{\sqrt{6 \times (185 + 82 \times \sqrt{5})}} \right)^{1/3} \times (5 + 3 \times \sqrt{5}) \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
\( V \) — Volume of Hexakis Icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the midsphere radius based on the volume of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the Hexakis Icosahedron. It helps in determining the sphere that touches all edges of the polyhedron, which is useful in various mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the volume of the Hexakis Icosahedron in cubic meters. The value must be positive and valid. The calculator will compute the midsphere radius using the provided formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a polyhedron with 120 triangular faces, 180 edges, and 62 vertices. It is a Catalan solid, the dual of the truncated dodecahedron.

Q2: Why is the midsphere radius important?
A: The midsphere radius helps in understanding the geometric properties of the polyhedron and is used in various applications including computer graphics, crystallography, and architectural design.

Q3: What units should be used for volume?
A: Volume should be entered in cubic meters (m³). Ensure consistent units for accurate results.

Q4: Are there limitations to this formula?
A: The formula is specific to the Hexakis Icosahedron and assumes ideal geometric conditions. It may not apply to modified or irregular polyhedra.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Hexakis Icosahedron. Other polyhedra have different formulas for calculating midsphere radius.