Dodecahedral Edge Length of Truncated Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ le(Dodecahedron) = \frac{\sqrt{5} \times 4 \times r_{m}}{5 + (3 \times \sqrt{5})} \]

Dodecahedral Edge Length of Truncated Dodecahedron:

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1. What is Dodecahedral Edge Length of Truncated Dodecahedron?

The Dodecahedral Edge Length of Truncated Dodecahedron is the length of any edge of the larger dodecahedron from which the corners are cut to form the Truncated Dodecahedron. It represents the original edge length before truncation.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ le(Dodecahedron) = \frac{\sqrt{5} \times 4 \times r_{m}}{5 + (3 \times \sqrt{5})} \]

Where:

\( le(Dodecahedron) \) — Dodecahedral Edge Length of Truncated Dodecahedron (m)
\( r_{m} \) — Midsphere Radius of Truncated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the original dodecahedron edge length based on the midsphere radius of the truncated dodecahedron, using the mathematical relationship between these geometric properties.

3. Importance of Dodecahedral Edge Length Calculation

Details: Calculating the original dodecahedral edge length is crucial for understanding the geometric properties of truncated polyhedra, architectural design, crystallography, and mathematical modeling of three-dimensional structures.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding dodecahedral edge length.

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: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the polyhedron.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the relationship between the dodecahedral edge length and midsphere radius of a truncated dodecahedron.

Q4: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, molecular structures, and mathematical research involving polyhedral geometry.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy of the result depends on the precision of the input value.