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Surface To Volume Ratio Of Pentakis Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ RA/V = \frac{\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}}{23 + 11 \times \sqrt{5}}} \times \sqrt{\frac{15 \times \sqrt{413 + 162 \times \sqrt{5}}}{19 \times TSA}} \]

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

The Surface to Volume Ratio of Pentakis Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of a Pentakis Dodecahedron. It indicates how much surface area is available per unit volume of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ RA/V = \frac{\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}}{23 + 11 \times \sqrt{5}}} \times \sqrt{\frac{15 \times \sqrt{413 + 162 \times \sqrt{5}}}{19 \times TSA}} \]

Where:

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

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and physics. For polyhedra like the Pentakis Dodecahedron, this ratio helps understand properties related to surface area exposure relative to volume, which can be crucial in applications involving heat transfer, catalysis, and other surface-dependent phenomena.

4. Using the Calculator

Tips: Enter the total surface area of the Pentakis Dodecahedron in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding surface to volume ratio.

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

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is important in many scientific and engineering applications as it affects properties like heat dissipation, chemical reactivity, and mechanical strength.

Q3: What units should I use for the input?
A: The total surface area should be entered in square meters (m²), and the result will be in inverse meters (m⁻¹).

Q4: Can this calculator handle very large or very small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this formula specific to Pentakis Dodecahedron?
A: Yes, this formula is specifically derived for calculating the surface to volume ratio of a Pentakis Dodecahedron given its total surface area.

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