Leg Length Of Pentakis Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ l_{Leg} = \frac{3}{38} \cdot (9+\sqrt{5}) \cdot \frac{\frac{76}{19}}{\frac{A_s}{V}} \cdot \frac{\sqrt{413+162\sqrt{5}}}{23+11\sqrt{5}} \]

Leg Length of Pentakis Dodecahedron:

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1. What is Leg Length of Pentakis Dodecahedron?

The Leg Length of Pentakis Dodecahedron refers to the length of the equal sides of the isosceles triangular faces that make up this polyhedron. It is a key geometric parameter in understanding the structure and properties of the Pentakis Dodecahedron.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ l_{Leg} = \frac{3}{38} \cdot (9+\sqrt{5}) \cdot \frac{\frac{76}{19}}{\frac{A_s}{V}} \cdot \frac{\sqrt{413+162\sqrt{5}}}{23+11\sqrt{5}} \]

Where:

\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( \frac{A_s}{V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

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

3. Importance of Leg Length Calculation

Details: Calculating the leg length is essential for understanding the geometric properties, surface area, volume, and other characteristics of the Pentakis Dodecahedron. It is particularly useful in crystallography, materials science, and geometric modeling applications.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and non-zero. The calculator will compute the corresponding leg length of the Pentakis Dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that can be obtained by adding a pyramid to each face of a regular dodecahedron. It has 60 isosceles triangular faces.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is a critical parameter that influences various physical and chemical properties, particularly in materials science and nanotechnology applications.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size and geometry of the Pentakis Dodecahedron. Smaller polyhedra typically have higher surface to volume ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Pentakis Dodecahedron. Other polyhedra have different geometric relationships and require different formulas.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy of the result depends on the precision of the input value and the computational precision of the system.