Insphere Radius of Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{25 + 11\sqrt{5}}{10}} \times 2r_m}{3 + \sqrt{5}} \]

Insphere Radius of Dodecahedron:

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1. What is Insphere Radius of Dodecahedron?

The Insphere Radius of a Dodecahedron is the radius of the largest sphere that can be contained within a regular dodecahedron such that the sphere touches all the faces of the dodecahedron tangentially.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{25 + 11\sqrt{5}}{10}} \times 2r_m}{3 + \sqrt{5}} \]

Where:

\( r_i \) — Insphere Radius of Dodecahedron (m)
\( r_m \) — Midsphere Radius of Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the insphere radius and midsphere radius of a regular dodecahedron based on 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 midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the difference between insphere radius and midsphere radius?
A: The insphere radius touches all faces, while the midsphere radius touches all edges of the dodecahedron.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons.

Q4: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular structure analysis, geodesic dome design, and various engineering applications.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons, with accuracy limited only by the precision of the input values and computational rounding.