Surface to Volume Ratio of Icosidodecahedron given Midsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{6 \times (5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}})}{\frac{2r_m}{\sqrt{5 + 2\sqrt{5}}} \times (45 + 17\sqrt{5})} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Icosidodecahedron?

The Surface to Volume Ratio of an Icosidodecahedron is a measure that compares the total surface area to the volume of this Archimedean solid. It's an important geometric property that indicates how much surface area is available per unit volume.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{6 \times (5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}})}{\frac{2r_m}{\sqrt{5 + 2\sqrt{5}}} \times (45 + 17\sqrt{5})} \]

Where:

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

Explanation: This formula calculates the ratio of surface area to volume based on the midsphere radius, which is the radius of the sphere tangent to all edges of the Icosidodecahedron.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in various applications including material science, chemistry, and physics. For geometric solids, it helps understand properties like heat transfer, diffusion rates, and structural efficiency.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio in reciprocal meters (m⁻¹).

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosidodecahedron?
A: An Icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 identical vertices, and 60 identical edges.

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: This ratio is important in many scientific fields as it affects properties like reaction rates, heat dissipation, and structural strength.

Q4: What are typical values for this ratio?
A: The ratio depends on the size of the Icosidodecahedron. For a given shape, the ratio decreases as the size increases.

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