Surface To Volume Ratio Of Pentakis Dodecahedron Given Insphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{\left(\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}\right)}{23 + 11 \times \sqrt{5}} \times \frac{3 \times \sqrt{\frac{81 + 35 \times \sqrt{5}}{218}}}{2 \times \text{Insphere Radius}} \]

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 relates the total surface area of the polyhedron to its volume. It is an important parameter in materials science, chemistry, and physics where surface effects play a significant role.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

Insphere Radius - The radius of the sphere that is contained by the Pentakis Dodecahedron
√5 - Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the ratio of surface area to volume based on the insphere radius of the Pentakis Dodecahedron.

3. Importance of Surface to Volume Ratio Calculation

Details: Surface to volume ratio is crucial in various scientific fields. Higher ratios indicate more surface area relative to volume, which is important in processes like catalysis, heat transfer, and chemical reactions.

4. Using the Calculator

Tips: Enter the insphere 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 insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the Pentakis Dodecahedron, touching all its faces.

Q3: Why is surface to volume ratio important?
A: It determines how efficiently a shape can exchange matter or energy with its environment, which is critical in many physical and chemical processes.

Q4: What units are used for the result?
A: The surface to volume ratio is expressed in reciprocal meters (m⁻¹), which is the standard SI unit for this measurement.

Q5: 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 surface to volume ratio.