Circumsphere Radius of Truncated Icosahedron Given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}}}{4} \times \left( \frac{4V}{125 + 43\sqrt{5}} \right)^{1/3} \]

Circumsphere Radius (r_c):

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1. What is the Circumsphere Radius of a Truncated Icosahedron?

The circumsphere radius of a truncated icosahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices lie on the sphere's surface. This measurement is crucial for understanding the spatial properties of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{58 + 18\sqrt{5}}}{4} \times \left( \frac{4V}{125 + 43\sqrt{5}} \right)^{1/3} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of truncated icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the geometric properties of the truncated icosahedron, relating its volume to the radius of its circumscribed sphere.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the truncated icosahedron, which has applications in various fields including molecular modeling, architecture, and material science.

4. Using the Calculator

Tips: Enter the volume of the truncated icosahedron in cubic meters. The value must be positive and non-zero for accurate calculation.

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), best known as the shape of a soccer ball.

Q2: Why is this calculation important?
A: Understanding the circumsphere radius helps in spatial analysis, packaging problems, and molecular structure studies where this shape appears.

Q3: What units should I use?
A: The calculator uses meters for both input (volume) and output (radius). Ensure consistent units for accurate results.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated icosahedra. Other polyhedra have different relationships between volume and circumsphere radius.

Q5: What's the precision of the calculation?
A: The calculator provides results with 10 decimal places precision, suitable for most scientific and engineering applications.