Circumsphere Radius of Truncated Dodecahedron Given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{74 + 30\sqrt{5}}}{4} \times \left( \frac{12V}{5(99 + 47\sqrt{5})} \right)^{1/3} \]

Circumsphere Radius of Truncated Dodecahedron (r_c):

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1. What is the Circumsphere Radius of Truncated Dodecahedron?

The circumsphere radius of a truncated dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the truncated dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{74 + 30\sqrt{5}}}{4} \times \left( \frac{12V}{5(99 + 47\sqrt{5})} \right)^{1/3} \]

Where:

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

Explanation: This formula derives from the geometric properties of the truncated dodecahedron and establishes the relationship between its volume and circumsphere radius.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry, crystallography, and materials science for understanding the spatial dimensions and packing efficiency of truncated dodecahedral structures.

4. Using the Calculator

Tips: Enter the volume of the truncated dodecahedron in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by cutting the corners of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

Q2: Why is the circumsphere radius important?
A: The circumsphere radius helps determine the minimum container size needed to enclose the polyhedron and is crucial for spatial analysis and packing problems.

Q3: What units should I use for volume input?
A: The calculator expects volume in cubic meters (m³). If your volume is in different units, convert it to cubic meters first.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the truncated dodecahedron, providing precise results for any valid volume input.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated dodecahedra. Other polyhedra have different formulas for calculating circumsphere radius from volume.