Insphere Radius of Dodecahedron given Face Diagonal Calculator

Formula:

\[ r_i = \frac{\sqrt{\frac{25 + 11\sqrt{5}}{10}} \times d_{Face}}{1 + \sqrt{5}} \]

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 the dodecahedron such that the sphere touches all the faces of the polyhedron. It represents the distance from the center of the dodecahedron to the center of any face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{25 + 11\sqrt{5}}{10}} \times d_{Face}}{1 + \sqrt{5}} \]

Where:

\( r_i \) — Insphere Radius of Dodecahedron (m)
\( d_{Face} \) — Face Diagonal of Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the face diagonal measurement of a regular dodecahedron, utilizing the mathematical constant φ (phi) and its relationship with √5.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry, material science, and engineering applications where dodecahedral structures are used. It helps determine the maximum size of spherical objects that can fit perfectly inside a dodecahedral container.

4. Using the Calculator

Tips: Enter the face diagonal measurement of the dodecahedron 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. It is one of the five Platonic solids.

Q2: How is face diagonal different from space diagonal?
A: Face diagonal connects opposite corners within the same pentagonal face, while space diagonal connects opposite vertices through the interior of the dodecahedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, packaging optimization, and in the study of molecular structures with dodecahedral symmetry.

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 and all angles are equal.

Q5: What is the relationship between insphere radius and circumsphere radius?
A: For a regular dodecahedron, the circumsphere radius (radius of sphere containing the dodecahedron) is larger than the insphere radius, with a specific mathematical relationship between them.