Surface to Volume Ratio of Truncated Icosidodecahedron given Midsphere Radius Calculator

Formula Used:

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

Surface to Volume Ratio (SA:V):

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1. What is Surface to Volume Ratio of Truncated Icosidodecahedron?

The Surface to Volume Ratio (SA:V) of a Truncated Icosidodecahedron is the numerical ratio of its total surface area to its volume. It's an important geometric property that describes how much surface area the polyhedron has relative to its volume.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_m \) — Midsphere radius of the truncated icosidodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the surface to volume ratio based on the midsphere radius, incorporating the mathematical constants and geometric properties specific to the truncated icosidodecahedron.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and engineering. It helps determine properties like reactivity, heat transfer efficiency, and structural characteristics of geometric shapes.

4. Using the Calculator

Tips: Enter the midsphere radius in 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 hexagons, and 12 decagons), 180 edges, and 120 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which affects properties like diffusion rates, heat dissipation, and chemical reactivity.

Q4: What are typical values for this ratio?
A: The ratio depends on the size of the polyhedron. Smaller polyhedra generally have higher surface to volume ratios.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for truncated icosidodecahedra. Other polyhedra have different formulas for surface to volume ratio.