Insphere Radius of Pentakis Dodecahedron given Total Surface Area Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \sqrt{\frac{19 \times TSA}{15 \times \sqrt{413 + 162\sqrt{5}}}}} \]

Insphere Radius (rᵢ):

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1. What is the Insphere Radius of Pentakis Dodecahedron?

The Insphere Radius of a Pentakis Dodecahedron is the radius of the largest sphere that can be inscribed within the polyhedron such that it touches all faces tangentially. It represents the distance from the center to the innermost surface of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \sqrt{\frac{19 \times TSA}{15 \times \sqrt{413 + 162\sqrt{5}}}}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{5} \) — Square root of 5 (≈2.23607)

Explanation: This formula calculates the insphere radius based on the total surface area 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 internal spatial properties of the Pentakis Dodecahedron, including packing efficiency and volume relationships.

4. Using the Calculator

Tips: Enter the total surface area in square 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, featuring 60 isosceles triangular faces.

Q2: How is this formula derived?
A: The formula is derived from geometric relationships between the total surface area and the insphere radius of the Pentakis Dodecahedron, incorporating mathematical constants specific to this polyhedron.

Q3: What units should I use?
A: Use consistent units (preferably meters for length and square meters for area) to ensure accurate results.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric Pentakis Dodecahedron and may not account for manufacturing tolerances or material properties in real-world applications.

Q5: Can this be used for other polyhedra?
A: No, this specific formula applies only to the Pentakis Dodecahedron. Other polyhedra have different formulas for calculating insphere radius.