Insphere Radius of Icosahedron given Surface to Volume Ratio Calculator

Formula:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \frac{12 \cdot \sqrt{3}}{(3 + \sqrt{5}) \cdot \frac{R_A}{V}} \]

Insphere Radius (r_i):

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1. What is the Insphere Radius of Icosahedron?

The Insphere Radius of an Icosahedron is the radius of the sphere that is contained by the Icosahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \frac{12 \cdot \sqrt{3}}{(3 + \sqrt{5}) \cdot \frac{R_A}{V}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( \frac{R_A}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{3} \) — Square root of 3 (≈1.732)
\( \sqrt{5} \) — Square root of 5 (≈2.236)

Explanation: This formula calculates the insphere radius of a regular icosahedron based on its surface to volume ratio, using mathematical constants and geometric relationships specific to the icosahedron's structure.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry, material science, and various engineering applications where understanding the internal space and packing efficiency of icosahedral structures is required.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular icosahedron?
A: A regular icosahedron is a polyhedron with 20 equilateral triangular faces, 12 vertices, and 30 edges. It is one of the five Platonic solids.

Q2: How is surface to volume ratio defined for an icosahedron?
A: The surface to volume ratio is calculated by dividing the total surface area by the volume of the icosahedron.

Q3: What are typical applications of icosahedral structures?
A: Icosahedral structures are used in architecture, molecular modeling (such as viral capsids), geodesic domes, and various mathematical and geometric applications.

Q4: Are there limitations to this calculation?
A: This formula applies only to regular icosahedrons. For irregular or modified icosahedral shapes, different calculations would be required.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular icosahedrons, provided the input surface to volume ratio is accurate.