Surface to Volume Ratio of Pentakis Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{\left(\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}\right)}{23 + 11 \times \sqrt{5}} \times \frac{3 + \sqrt{5}}{4 \times \text{Midsphere Radius}} \]

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 is an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ \text{Surface to Volume Ratio} = \frac{\left(\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}\right)}{23 + 11 \times \sqrt{5}} \times \frac{3 + \sqrt{5}}{4 \times \text{Midsphere Radius}} \]

Where:

Midsphere Radius - The radius of the sphere that touches the midpoints of all edges of the Pentakis Dodecahedron
√5 - Square root of 5 (approximately 2.23607)

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

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and engineering. It helps determine properties like reactivity, heat transfer efficiency, and structural characteristics of three-dimensional shapes.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio in reciprocal meters (m⁻¹).

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: What is the significance of the midsphere radius?
A: The midsphere radius is the radius of the sphere that touches the midpoints of all edges of the polyhedron, providing important geometric information about the shape's proportions.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the Pentakis Dodecahedron. Smaller polyhedra have higher ratios, while larger ones have lower ratios.

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Pentakis Dodecahedron. Other polyhedra have different formulas for calculating surface to volume ratios.

Q5: What units should I use for the midsphere radius?
A: The calculator expects the midsphere radius in meters, but you can use any consistent unit system as long as you interpret the result in the corresponding reciprocal units.