Volume of Deltoidal Icositetrahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ V = \frac{2}{7} \times \sqrt{292 + (206 \times \sqrt{2})} \times \left( \frac{6}{SA:V} \times \sqrt{\frac{61 + (38 \times \sqrt{2})}{292 + (206 \times \sqrt{2})}} \right)^3 \]

Volume of Deltoidal Icositetrahedron:

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1. What is the Volume of Deltoidal Icositetrahedron?

The Deltoidal Icositetrahedron is a Catalan solid with 24 deltoid faces. Its volume represents the three-dimensional space enclosed by its surface, calculated based on its surface to volume ratio.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{2}{7} \times \sqrt{292 + (206 \times \sqrt{2})} \times \left( \frac{6}{SA:V} \times \sqrt{\frac{61 + (38 \times \sqrt{2})}{292 + (206 \times \sqrt{2})}} \right)^3 \]

Where:

\( V \) — Volume of Deltoidal Icositetrahedron (m³)
\( SA:V \) — Surface to Volume Ratio (1/m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula derives the volume from the surface to volume ratio using geometric properties specific to the Deltoidal Icositetrahedron.

3. Importance of Volume Calculation

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

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Icositetrahedron?
A: It's a Catalan solid with 24 deltoid (kite-shaped) faces, 26 vertices, and 48 edges.

Q2: What are typical SA:V values for this shape?
A: The surface to volume ratio depends on the specific dimensions, but generally ranges based on the size and proportions of the solid.

Q3: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Deltoidal Icositetrahedron due to its unique geometric properties.

Q4: What are the practical applications of this calculation?
A: Useful in crystallography, architectural design, and mathematical modeling where this specific polyhedron appears.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the derived formula, though numerical precision depends on implementation.