Total Surface Area Of Deltoidal Icositetrahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{6}{AV} \times \sqrt{\frac{61 + 38\sqrt{2}}{292 + 206\sqrt{2}}} \right)^2 \]

Total Surface Area of Deltoidal Icositetrahedron:

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1. What is the Deltoidal Icositetrahedron?

The Deltoidal Icositetrahedron is a Catalan solid with 24 deltoid (kite-shaped) faces. It is the dual polyhedron of the rhombicuboctahedron and has interesting geometric properties.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{6}{AV} \times \sqrt{\frac{61 + 38\sqrt{2}}{292 + 206\sqrt{2}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Deltoidal Icositetrahedron
\( AV \) — Surface to Volume Ratio of Deltoidal Icositetrahedron
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the total surface area based on the surface to volume ratio of the deltoidal icositetrahedron, using the mathematical relationship between these geometric properties.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area of geometric solids is important in various fields including mathematics, engineering, architecture, and materials science for understanding material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the surface to volume ratio (SA:V) of the deltoidal icositetrahedron. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a deltoidal icositetrahedron?
A: It's a Catalan solid with 24 kite-shaped faces, 48 edges, and 26 vertices. It's the dual of the rhombicuboctahedron.

Q2: What are the applications of this calculation?
A: This calculation is useful in geometric modeling, architectural design, material science, and any field requiring precise surface area measurements of complex polyhedra.

Q3: What units should I use for the input?
A: The surface to volume ratio should be in reciprocal meters (1/m), and the result will be in square meters (m²).

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal icositetrahedron. Other polyhedra have different geometric relationships.

Q5: What if I get an unexpected result?
A: Double-check your input value for the surface to volume ratio. Ensure it's a positive number and represents the correct geometric property of a deltoidal icositetrahedron.