Surface to Volume Ratio of Truncated Dodecahedron given Circumsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}})}{\frac{4 \times R_c}{\sqrt{74 + 30 \times \sqrt{5}}} \times (99 + 47 \times \sqrt{5})} \]

Surface to Volume Ratio:

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

The Surface to Volume Ratio of a Truncated Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of this polyhedron. It is an important parameter in materials science, chemistry, and physics for understanding surface properties relative to volume.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{12 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}})}{\frac{4 \times R_c}{\sqrt{74 + 30 \times \sqrt{5}}} \times (99 + 47 \times \sqrt{5})} \]

Where:

\( R_c \) — Circumsphere Radius of the Truncated Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the ratio of surface area to volume based on the circumsphere radius, incorporating the mathematical constants and relationships specific to the truncated dodecahedron geometry.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various scientific fields. In materials science, it affects reactivity and strength. In chemistry, it influences reaction rates. In physics, it relates to heat transfer and other surface-dependent phenomena.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and non-zero. The calculator will compute the surface to volume ratio based on the geometric properties of a truncated dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by truncating the vertices of a regular dodecahedron. It has 32 faces: 20 regular triangles and 12 regular decagons.

Q2: What does the Surface to Volume Ratio represent?
A: It represents how much surface area is available per unit volume, which is important for understanding how the shape interacts with its environment.

Q3: Why is the circumsphere radius used in this calculation?
A: The circumsphere radius is a fundamental geometric property that defines the size of the polyhedron and is used to derive other geometric properties.

Q4: What are typical values for this ratio?
A: The ratio depends on the size of the polyhedron. Smaller polyhedra generally have higher surface to volume ratios than larger ones of the same shape.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for truncated dodecahedra. Other polyhedra have different geometric relationships and require different formulas.