Surface To Volume Ratio Of Icosahedron Given Insphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{3}}{(3+\sqrt{5}) \times \left( \frac{12 \times \text{Insphere Radius}}{\sqrt{3}(3+\sqrt{5})} \right)} \]

Surface to Volume Ratio:

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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 regular polyhedron with 20 equilateral triangular faces. It's an important parameter in various scientific and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{3}}{(3+\sqrt{5}) \times \left( \frac{12 \times \text{Insphere Radius}}{\sqrt{3}(3+\sqrt{5})} \right)} \]

Where:

\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)
Insphere Radius — Radius of the sphere inscribed within the icosahedron

Explanation: This formula calculates the ratio of surface area to volume based on the insphere radius of a regular icosahedron.

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 reactivity, heat transfer, and diffusion rates. For icosahedral structures, this ratio is particularly important in nanotechnology and molecular modeling.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-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 is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the icosahedron, touching all faces.

Q3: Why is surface to volume ratio important?
A: This ratio is critical in many physical and chemical processes where surface interactions dominate, such as catalysis, heat transfer, and biological systems.

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

Q5: Can this formula be used for irregular icosahedra?
A: No, this formula is specifically for regular icosahedra where all faces are equilateral triangles and all vertices are equivalent.