Volume of Truncated Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{4r_m}{3(1 + \sqrt{5})} \right)^3 \]

Volume of Truncated Icosahedron:

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1. What is the Volume of Truncated Icosahedron?

The Volume of Truncated Icosahedron is the total quantity of three dimensional space enclosed by the surface of the Truncated Icosahedron. It is a polyhedron with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{4r_m}{3(1 + \sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Truncated Icosahedron (m³)
\( r_m \) — Midsphere Radius of Truncated Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the volume of a truncated icosahedron based on its midsphere radius, incorporating the mathematical constant √5 which is characteristic of this Archimedean solid.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in various fields including mathematics, engineering, architecture, and materials science. For the truncated icosahedron (which resembles a soccer ball), volume calculations are particularly relevant in nanotechnology and fullerene research.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), 90 edges, and 60 vertices. It's the shape of a soccer ball and C₆₀ fullerene molecules.

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: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters first, then convert the result back to your desired unit.

Q4: How accurate is the calculation?
A: The calculation uses precise mathematical formulas and PHP's built-in mathematical functions, providing results accurate to 6 decimal places.

Q5: What are some real-world applications of this calculation?
A: This calculation is used in materials science (particularly in fullerene research), architectural design, game development, and any field dealing with geometric modeling of complex polyhedra.