Edge Length of Truncated Icosidodecahedron given Volume Calculator

Formula Used:

\[ l_e = \left( \frac{V}{5 \times (19 + 10 \times \sqrt{5})} \right)^{1/3} \]

Edge Length of Truncated Icosidodecahedron:

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1. What is the Edge Length of Truncated Icosidodecahedron?

The edge length of a truncated icosidodecahedron is the length of any edge of this Archimedean solid. It is a key geometric parameter used in various mathematical and engineering calculations involving this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \left( \frac{V}{5 \times (19 + 10 \times \sqrt{5})} \right)^{1/3} \]

Where:

\( l_e \) — Edge length of truncated icosidodecahedron (meters)
\( V \) — Volume of truncated icosidodecahedron (cubic meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the mathematical relationship between the volume and edge length of a truncated icosidodecahedron, incorporating the mathematical constant φ (phi) through the square root of 5.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from volume is essential for geometric modeling, architectural design, material estimation, and various engineering applications involving this specific polyhedral shape.

4. Using the Calculator

Tips: Enter the volume of the truncated icosidodecahedron in cubic meters. The value must be positive and non-zero. The calculator will compute the corresponding edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosidodecahedron?
A: A truncated icosidodecahedron is an Archimedean solid with 62 faces (30 squares, 20 hexagons, and 12 decagons), 180 edges, and 120 vertices.

Q2: Why is the square root of 5 in the formula?
A: The square root of 5 appears because it's related to the golden ratio φ = (1+√5)/2, which is fundamental to the geometry of regular and semi-regular polyhedra.

Q3: Can this formula be used for scaling models?
A: Yes, this formula is particularly useful for scaling models while maintaining geometric proportions between volume and edge length.

Q4: What are typical applications of this calculation?
A: Applications include architectural design, molecular modeling, game development, and mathematical education involving polyhedral geometry.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect truncated icosidodecahedra. The accuracy depends on the precision of the input volume value.