Base Length of Pentakis Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ l_{Base} = \frac{\frac{76}{19} \times \sqrt{413 + (162 \times \sqrt{5})}}{R_{A/V} \times (23 + (11 \times \sqrt{5}))} \]

Base Length of Pentakis Dodecahedron:

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1. What is Base Length of Pentakis Dodecahedron?

The Base Length of Pentakis Dodecahedron is the length of the base of the isosceles triangular face of a Pentakis Dodecahedron. It is a crucial geometric parameter that helps define the shape and proportions of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ l_{Base} = \frac{\frac{76}{19} \times \sqrt{413 + (162 \times \sqrt{5})}}{R_{A/V} \times (23 + (11 \times \sqrt{5}))} \]

Where:

\( l_{Base} \) — Base Length of Pentakis Dodecahedron (meters)
\( R_{A/V} \) — Surface to Volume Ratio (1/meter)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the base 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 Base Length Calculation

Details: Calculating the base length is essential for geometric analysis, 3D modeling, and understanding the spatial properties of Pentakis Dodecahedrons in various applications including crystallography, architecture, and mathematical research.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/meter units. 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 can be seen as a dodecahedron with a pyramid on each face, creating 60 isosceles triangular faces.

Q2: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies depending on the size and proportions of the polyhedron, but typically ranges from 0.1 to 10 1/m for most practical applications.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Pentakis Dodecahedron, assuming precise input values.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to the Pentakis Dodecahedron due to its unique geometric properties.

Q5: What units should I use for the result?
A: The result is in meters, but you can convert it to other length units as needed for your specific application.