Surface to Volume Ratio of Icosahedron given Circumsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio of Icosahedron} = \frac{12\sqrt{3}}{(3+\sqrt{5}) \cdot \left(\frac{4 \cdot \text{Circumsphere Radius of Icosahedron}}{\sqrt{10+2\sqrt{5}}}\right)} \]

Surface to Volume Ratio of Icosahedron:

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1. What is Surface to Volume Ratio of Icosahedron?

The Surface to Volume Ratio of an Icosahedron is a geometric measurement that compares the total surface area to the volume of this twenty-faced polyhedron. It's an important parameter in various scientific and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{3}}{(3+\sqrt{5}) \cdot \left(\frac{4 \cdot r_c}{\sqrt{10+2\sqrt{5}}}\right)} \]

Where:

\( r_c \) — Circumsphere Radius of Icosahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the ratio between the total surface area and volume of a regular icosahedron based on its circumsphere radius.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in materials science, chemistry, and physics as it affects properties like reaction rates, heat transfer, and structural efficiency. For icosahedral structures, this ratio is particularly important in nanotechnology and molecular modeling.

4. Using the Calculator

Tips: Enter the circumsphere 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 a regular icosahedron?
A: A regular icosahedron is a convex polyhedron with 20 identical equilateral triangular faces, 30 edges, and 12 vertices.

Q2: What does the circumsphere radius represent?
A: The circumsphere radius is the radius of the sphere that passes through all vertices of the icosahedron.

Q3: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which is critical for processes involving surface interactions like catalysis, diffusion, and heat exchange.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before calculation.

Q5: What are typical values for this ratio?
A: The surface to volume ratio depends on the size of the icosahedron. Smaller icosahedra have higher ratios, while larger ones have lower ratios.