Volume of Truncated Icosahedron given Circumsphere Radius Calculator

Formula Used:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{4r_c}{\sqrt{58 + 18\sqrt{5}}} \right)^3 \]

Volume of Truncated Icosahedron:

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1. What is Volume of Truncated Icosahedron?

The volume of a truncated icosahedron is the total quantity of three dimensional space enclosed by the surface of this Archimedean solid. A truncated icosahedron 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:

\[ V = \frac{125 + 43\sqrt{5}}{4} \times \left( \frac{4r_c}{\sqrt{58 + 18\sqrt{5}}} \right)^3 \]

Where:

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

Explanation: This formula calculates the volume of a truncated icosahedron based on the radius of its circumscribed sphere, using the mathematical relationship between these geometric properties.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and materials science. For truncated icosahedrons specifically, this is important in understanding molecular structures (like fullerenes), architectural designs, and various engineering applications.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and non-zero. 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 obtained by truncating the vertices of a regular icosahedron, resulting in 12 pentagonal and 20 hexagonal faces.

Q2: Where are truncated icosahedrons found in nature?
A: The molecular structure of buckminsterfullerene (C60) is a truncated icosahedron, and this shape is also found in some viruses and architectural designs.

Q3: What are the units for the calculation?
A: The calculator uses meters for input (circumsphere radius) and outputs cubic meters for volume. Ensure consistent units for accurate results.

Q4: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula. The accuracy depends on the precision of the input value.

Q5: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to truncated icosahedrons. Other polyhedrons have different volume formulas based on their unique geometric properties.