Midsphere Radius of Truncated Icosahedron given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{3(1+\sqrt{5})}{4} \times \sqrt{\frac{TSA}{3((10\sqrt{3})+\sqrt{25+(10\sqrt{5})})}} \]

Midsphere Radius of Truncated Icosahedron:

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1. What is the Midsphere Radius of Truncated Icosahedron?

The Midsphere Radius of a Truncated Icosahedron is the radius of the sphere that touches all the edges of the polyhedron. For a Truncated Icosahedron (a shape with 12 pentagonal faces and 20 hexagonal faces), this sphere is tangent to all its edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3(1+\sqrt{5})}{4} \times \sqrt{\frac{TSA}{3((10\sqrt{3})+\sqrt{25+(10\sqrt{5})})}} \]

Where:

\( r_m \) — Midsphere Radius of Truncated Icosahedron (m)
\( TSA \) — Total Surface Area of Truncated Icosahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the midsphere radius based on the total surface area of the truncated icosahedron, incorporating mathematical constants related to its geometric properties.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and materials science for understanding the spatial properties and packing characteristics of truncated icosahedral structures, which include famous molecules like buckminsterfullerene (C60).

4. Using the Calculator

Tips: Enter the total surface area of the truncated icosahedron in square meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

Q2: Why is this shape significant?
A: This shape is famous as the molecular structure of buckminsterfullerene (C60) and is also used in soccer ball designs.

Q3: What are typical values for the midsphere radius?
A: The midsphere radius depends on the size of the polyhedron. For a standard truncated icosahedron with edge length a, the midsphere radius is approximately 2.478a.

Q4: Can this calculator handle different units?
A: The calculator expects input in square meters and returns results in meters. For other units, convert your measurements accordingly before calculation.

Q5: What is the precision of the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most geometric and engineering applications.