Insphere Radius of Dodecahedron given Space Diagonal Calculator

Formula Used:

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

Insphere Radius of Dodecahedron:

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1. What is Insphere Radius of Dodecahedron?

The Insphere Radius of Dodecahedron is the radius of the sphere that is contained by the Dodecahedron in such a way that all the faces just touch the sphere. It represents the maximum sphere that can fit inside the dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_i \) — Insphere Radius of Dodecahedron (m)
\( d_{Space} \) — Space Diagonal of Dodecahedron (m)

Explanation: This formula calculates the insphere radius based on the space diagonal measurement of a regular dodecahedron, using mathematical constants related to its geometric properties.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling for understanding the spatial relationships within a dodecahedron and for applications in crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the space diagonal measurement in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: How is space diagonal different from face diagonal?
A: Space diagonal connects two vertices that are not on the same face, while face diagonal connects two non-adjacent vertices on the same face.

Q3: What are the practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, game development, and architectural design where dodecahedral structures are employed.

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

Q5: What is the relationship between insphere radius and circumsphere radius?
A: The insphere radius is always smaller than the circumsphere radius in a regular dodecahedron, with a specific mathematical ratio between them.