Surface to Volume Ratio of Truncated Dodecahedron given Total Surface Area Calculator

Formula Used:

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

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 relates the total surface area to the volume of this polyhedron. A truncated dodecahedron is an Archimedean solid created by truncating the vertices of a regular dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{RA/V} = \frac{12 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}})}{\sqrt{\frac{TSA}{5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}})} \times (99 + 47 \times \sqrt{5})}} \]

Where:

\( RA/V \) — Surface to Volume Ratio (m⁻¹)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the surface area to volume ratio based on the given total surface area of the truncated dodecahedron.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and physics. It affects properties like heat transfer, chemical reactivity, and mechanical strength of materials with this geometric shape.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid with 32 faces - 20 regular triangles and 12 regular decagons.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which is crucial for processes like heat exchange, diffusion, and chemical reactions.

Q3: What units should I use for input?
A: Use square meters for surface area. The calculator will output the ratio in meters⁻¹.

Q4: Can this calculator handle very large or small values?
A: Yes, but extremely large or small values may be limited by floating-point precision in PHP.

Q5: Is this formula applicable to all polyhedra?
A: No, this specific formula is derived specifically for the truncated dodecahedron geometry.