Insphere Radius of Pentakis Dodecahedron given Leg Length Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{38 \times l_{Leg}}{3 \times (9 + \sqrt{5})} \]

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 touching all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{38 \times l_{Leg}}{3 \times (9 + \sqrt{5})} \]

Where:

\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the insphere radius based on the leg length 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 characteristics.

4. Using the Calculator

Tips: Enter the Leg Length of Pentakis Dodecahedron in meters. The value must be positive and greater than zero to produce a valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that can be seen as a dodecahedron with a pyramid on each face, creating a polyhedron with 60 identical isosceles triangular faces.

Q2: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron and touches all faces, while the circumsphere radius is the radius of the smallest sphere that contains the polyhedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and any field that involves geometric analysis of complex polyhedral structures.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for the Pentakis Dodecahedron and assumes perfect geometric proportions. It may not apply to distorted or irregular variations of the shape.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, or convert the result from meters to your desired unit.