Surface to Volume Ratio of Great Dodecahedron given Volume Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{15\sqrt{5-2\sqrt{5}}}{\frac{5}{4}(\sqrt{5}-1)} \times \left( \frac{5(\sqrt{5}-1)}{4V} \right)^{\frac{1}{3}} \]

Surface to Volume Ratio:

Unit Converter ▲

Unit Converter ▼

From:	To:

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

The Surface to Volume Ratio of a Great Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of this particular polyhedron. It is an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{15\sqrt{5-2\sqrt{5}}}{\frac{5}{4}(\sqrt{5}-1)} \times \left( \frac{5(\sqrt{5}-1)}{4V} \right)^{\frac{1}{3}} \]

Where:

\( V \) — Volume of the Great Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the surface to volume ratio based on the given volume of the Great 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, heat transfer, and chemical reactions where the relationship between surface area and volume affects physical properties and behaviors.

4. Using the Calculator

Tips: Enter the volume of the Great Dodecahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is one of the Kepler-Poinsot polyhedra, consisting of 12 pentagonal faces that intersect each other.

Q2: What units should I use for volume?
A: The calculator expects volume in cubic meters (m³). If you have volume in other units, convert it to cubic meters first.

Q3: Can this calculator handle very large or very small volumes?
A: The calculator can handle a wide range of volume values, but extremely large or small values may affect computational precision.

Q4: What is the typical range of surface to volume ratios for Great Dodecahedrons?
A: The ratio varies inversely with volume - larger volumes generally yield smaller surface to volume ratios, and vice versa.

Q5: Are there practical applications of this calculation?
A: Yes, this calculation is useful in geometric modeling, architectural design, and materials science where the properties of polyhedral structures are important.