Insphere Radius of Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \sqrt{\frac{25+11\sqrt{5}}{10}} \times \frac{6\sqrt{25+10\sqrt{5}}}{R_{A/V} \times (15+7\sqrt{5})} \]

Insphere Radius (r_i):

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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 to any face of the dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \sqrt{\frac{25+11\sqrt{5}}{10}} \times \frac{6\sqrt{25+10\sqrt{5}}}{R_{A/V} \times (15+7\sqrt{5})} \]

Where:

\( r_i \) — Insphere Radius (m)
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the surface to volume ratio of a regular dodecahedron, incorporating the mathematical constant √5 which is fundamental to pentagonal geometry.

3. Importance of Insphere Radius Calculation

Details: The insphere radius is crucial in geometry and materials science for determining the maximum size of spherical objects that can fit perfectly within a dodecahedral structure, and for understanding the packing efficiency in various applications.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. 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 regular pentagonal faces, 20 vertices, and 30 edges.

Q2: How is surface to volume ratio defined for a dodecahedron?
A: The surface to volume ratio is the total surface area of the dodecahedron divided by its volume, expressed in m⁻¹.

Q3: What are typical values for dodecahedron surface to volume ratio?
A: The ratio depends on the size of the dodecahedron. For a regular dodecahedron with edge length 'a', the ratio is approximately 2.694/a.

Q4: Why does the formula contain √5?
A: The constant √5 appears naturally in the geometry of pentagons and dodecahedrons due to their golden ratio relationships.

Q5: Can this calculator be used for irregular dodecahedrons?
A: No, this formula is specifically designed for regular dodecahedrons where all faces are identical regular pentagons.