Space Diagonal of Dodecahedron given Insphere Radius Calculator

Formula Used:

\[ d_{Space} = \sqrt{3} \times (1 + \sqrt{5}) \times \frac{r_i}{\sqrt{\frac{25 + 11\sqrt{5}}{10}}} \]

Space Diagonal of Dodecahedron:

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1. What is Space Diagonal of Dodecahedron?

The Space Diagonal of a Dodecahedron is the line connecting two vertices that are not on the same face of the Dodecahedron. It passes through the interior of the solid and represents the longest distance between any two vertices of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_{Space} = \sqrt{3} \times (1 + \sqrt{5}) \times \frac{r_i}{\sqrt{\frac{25 + 11\sqrt{5}}{10}}} \]

Where:

\( d_{Space} \) — Space Diagonal of Dodecahedron (m)
\( r_i \) — Insphere Radius of Dodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the space diagonal of a regular dodecahedron based on its insphere radius, using mathematical constants and geometric relationships specific to this Platonic solid.

3. Importance of Space Diagonal Calculation

Details: Calculating the space diagonal is important in geometry, architecture, and engineering for understanding the spatial dimensions and proportions of dodecahedral structures. It helps in determining the maximum internal distance within the solid.

4. Using the Calculator

Tips: Enter the insphere radius of the dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding space diagonal.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular dodecahedron?
A: A regular dodecahedron is a three-dimensional Platonic solid with 12 identical regular pentagonal faces, 20 vertices, and 30 edges.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the dodecahedron, touching all faces.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular dodecahedron, using precise geometric relationships.

Q4: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons.

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and mathematical research involving polyhedral geometry.