Volume of Deltoidal Hexecontahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{\frac{9}{45} \cdot \sqrt{10(157 + 31\sqrt{5})}}{AV \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^3 \]

Volume of Deltoidal Hexecontahedron:

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1. What is the Volume of Deltoidal Hexecontahedron?

The Deltoidal Hexecontahedron is a Catalan solid with 60 deltoid faces. Its volume represents the amount of three-dimensional space enclosed by this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{\frac{9}{45} \cdot \sqrt{10(157 + 31\sqrt{5})}}{AV \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}}} \right)^3 \]

Where:

\( V \) — Volume of Deltoidal Hexecontahedron (m³)
\( AV \) — Surface to Volume Ratio of Deltoidal Hexecontahedron (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume based on the surface to volume ratio of the deltoidal hexecontahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is essential in various fields including mathematics, engineering, architecture, and material science for understanding spatial properties and capacity.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: It's a Catalan solid with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices.

Q2: What units should I use for the input?
A: The surface to volume ratio should be entered in reciprocal meters (1/m).

Q3: What is the typical range of values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the polyhedron, with smaller polyhedra having larger ratios.

Q4: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of positive values, but extremely small values may approach computational limits.

Q5: Are there any limitations to this calculation?
A: The formula assumes a perfect geometric shape and may not account for real-world imperfections or variations.