Total Surface Area of Pentakis Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162\sqrt{5}} \times \left(\frac{\frac{76}{19}}{RA/V}\right)^2 \times \left(\frac{413 + 162\sqrt{5}}{23 + 11\sqrt{5}}\right)^2 \]

Total Surface Area of Pentakis Dodecahedron:

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1. What is the Total Surface Area of Pentakis Dodecahedron?

The Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. Its total surface area represents the sum of the areas of all its triangular faces, providing crucial information about its geometric properties and spatial characteristics.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162\sqrt{5}} \times \left(\frac{\frac{76}{19}}{RA/V}\right)^2 \times \left(\frac{413 + 162\sqrt{5}}{23 + 11\sqrt{5}}\right)^2 \]

Where:

\( TSA \) — Total Surface Area of Pentakis Dodecahedron (m²)
\( RA/V \) — Surface to Volume Ratio of Pentakis Dodecahedron (m⁻¹)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This complex formula derives from the geometric properties of the Pentakis Dodecahedron and its relationship between surface area and volume ratio.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is essential for various applications including material science, architectural design, chemical reactivity studies, and understanding the physical properties of this specific polyhedral structure.

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 Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid with 60 isosceles triangular faces, 90 edges, and 32 vertices, formed by placing a pyramid on each face of a regular dodecahedron.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Pentakis Dodecahedron and the mathematical relationships between its surface area, volume, and their ratio.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the specific dimensions of the Pentakis Dodecahedron, but typically ranges based on the geometric configuration.

Q4: Can this calculator handle very large or small values?
A: The calculator is designed to handle a wide range of positive values, though extremely large or small numbers may have precision limitations.

Q5: What units should I use?
A: Use consistent units - surface to volume ratio in m⁻¹ will yield surface area in m². Other consistent unit systems can be used as well.