Insphere Radius of Pentakis Dodecahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \left( \frac{76V}{15(23 + 11\sqrt{5})} \right)^{\frac{1}{3}} \]

Insphere Radius of Pentakis Dodecahedron:

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1. What is the Insphere Radius of Pentakis Dodecahedron?

The Insphere Radius of Pentakis Dodecahedron is the radius of the sphere that is contained by the Pentakis Dodecahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \left( \frac{76V}{15(23 + 11\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( V \) — Volume of Pentakis Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the volume of the Pentakis Dodecahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the spatial properties of the Pentakis Dodecahedron, including its packing efficiency and internal volume distribution.

4. Using the Calculator

Tips: Enter the volume of the Pentakis Dodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: How is this formula derived?
A: The formula is derived from geometric relationships and mathematical properties specific to the Pentakis Dodecahedron, using volume as the primary input parameter.

Q3: What are typical values for the insphere radius?
A: The insphere radius varies depending on the volume of the polyhedron. For a unit volume Pentakis Dodecahedron, the insphere radius is approximately 0.35 units.

Q4: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, though extremely large or small values may be limited by PHP's floating-point precision.

Q5: What are practical applications of this calculation?
A: This calculation is useful in crystallography, nanotechnology, and architectural design where Pentakis Dodecahedron structures are employed.