Midsphere Radius of Truncated Icosidodecahedron Given Volume Calculator

Formula Used:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}}}{2} \times \left( \frac{V}{5(19 + 10\sqrt{5})} \right)^{\frac{1}{3}} \]

Midsphere Radius of Truncated Icosidodecahedron:

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1. What is Midsphere Radius of Truncated Icosidodecahedron?

The Midsphere Radius of a Truncated Icosidodecahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. It represents the sphere that fits perfectly within the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}}}{2} \times \left( \frac{V}{5(19 + 10\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

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

Explanation: This formula derives the midsphere radius from the volume of the truncated icosidodecahedron using geometric relationships and mathematical constants 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 truncated icosidodecahedron and its relationship with inscribed spheres.

4. Using the Calculator

Tips: Enter the volume of the truncated icosidodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Icosidodecahedron?
A: A truncated icosidodecahedron is an Archimedean solid with 62 faces (30 squares, 20 regular hexagons, and 12 regular decagons), 180 edges, and 120 vertices.

Q2: How is the midsphere different from the insphere?
A: The midsphere is tangent to all edges, while the insphere is tangent to all faces of the polyhedron.

Q3: What are the applications of this calculation?
A: This calculation is used in mathematical geometry, 3D modeling, architectural design, and understanding the properties of complex polyhedra.

Q4: Are there limitations to this formula?
A: This formula is specific to the truncated icosidodecahedron and assumes perfect geometric proportions. It may not apply to distorted or irregular versions of the shape.

Q5: Can this calculator handle very large or small volumes?
A: Yes, the calculator can handle a wide range of volume values, though extremely large or small values may be limited by PHP's floating-point precision.