Surface To Volume Ratio Of Pentakis Dodecahedron Calculator

Surface To Volume Ratio Of Pentakis Dodecahedron Formula:

\[ \text{Surface to Volume Ratio} = \frac{\frac{76}{19} \times \sqrt{413 + (162 \times \sqrt{5})}}{l_{\text{Base}} \times (23 + (11 \times \sqrt{5}))} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Pentakis Dodecahedron?

The Surface to Volume Ratio of a Pentakis Dodecahedron is a geometric property that represents the relationship between the total surface area and the total volume of this polyhedron. It's an important measurement in materials science and geometry that indicates how much surface area is available per unit volume.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{\frac{76}{19} \times \sqrt{413 + (162 \times \sqrt{5})}}{l_{\text{Base}} \times (23 + (11 \times \sqrt{5}))} \]

Where:

\( l_{\text{Base}} \) — Base Length of Pentakis Dodecahedron (in meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the ratio of surface area to volume for a Pentakis Dodecahedron based on its base length. The result is expressed in reciprocal meters (m⁻¹).

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various scientific and engineering applications, including heat transfer analysis, chemical reactivity studies, material science, and nanotechnology where surface effects dominate material properties.

4. Using the Calculator

Tips: Enter the base length of the Pentakis Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio automatically.

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. It has 60 isosceles triangular faces, 90 edges, and 32 vertices.

Q2: What units are used for the calculation?
A: The base length should be in meters, and the resulting surface to volume ratio will be in reciprocal meters (m⁻¹).

Q3: How does the base length affect the surface to volume ratio?
A: The surface to volume ratio is inversely proportional to the base length. As the size increases, the ratio decreases.

Q4: What are typical values for this ratio?
A: The ratio depends entirely on the size of the Pentakis Dodecahedron. Smaller objects have higher surface to volume ratios.

Q5: Where is this calculation used in practical applications?
A: This calculation is important in materials science, nanotechnology, chemical engineering, and any field where surface effects relative to volume are significant.