Midsphere Radius of Truncated Icosidodecahedron given Surface to Volume Ratio Calculator

Midsphere Radius of Truncated Icosidodecahedron Formula:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}} \cdot 3(1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}{SA:V \cdot (19 + 10\sqrt{5})} \]

Midsphere Radius of Truncated Icosidodecahedron:

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1. What is Midsphere Radius of Truncated Icosidodecahedron?

The Midsphere Radius of Truncated Icosidodecahedron is the radius of the sphere for which all the edges of the Truncated Icosidodecahedron become a tangent line on that sphere. It represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}} \cdot 3(1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}{SA:V \cdot (19 + 10\sqrt{5})} \]

Where:

\( r_m \) — Midsphere Radius of Truncated Icosidodecahedron (m)
\( SA:V \) — Surface to Volume Ratio of Truncated Icosidodecahedron (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the midsphere radius based on the surface to volume ratio of the truncated icosidodecahedron, incorporating various mathematical constants and operations.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling as it helps understand the spatial properties and proportions of the truncated icosidodecahedron, which is an Archimedean solid with complex geometric properties.

4. Using the Calculator

Tips: Enter the surface to volume ratio (SA:V) of the truncated icosidodecahedron in 1/m. 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: What does the midsphere represent?
A: The midsphere is the sphere that is tangent to all edges of the polyhedron, touching each edge at its midpoint.

Q3: How is surface to volume ratio related to midsphere radius?
A: The surface to volume ratio provides information about the compactness of the shape, which directly influences the midsphere radius calculation.

Q4: What are typical values for SA:V ratio?
A: The surface to volume ratio depends on the size of the polyhedron. For a unit truncated icosidodecahedron, the SA:V ratio is approximately 0.1-0.2 1/m.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the truncated icosidodecahedron only. Other polyhedra have different formulas for calculating their midsphere radii.