Total Surface Area Of Deltoidal Icositetrahedron Given Volume Calculator

Formula Used:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{7V}{2\sqrt{292 + 206\sqrt{2}}} \right)^{\frac{2}{3}} \]

Total Surface Area of Deltoidal Icositetrahedron:

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1. What is the Total Surface Area of Deltoidal Icositetrahedron?

The Deltoidal Icositetrahedron is a Catalan solid with 24 deltoid faces. The total surface area represents the sum of the areas of all these deltoid faces that cover the surface of this three-dimensional geometric shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{7V}{2\sqrt{292 + 206\sqrt{2}}} \right)^{\frac{2}{3}} \]

Where:

\( TSA \) — Total Surface Area of Deltoidal Icositetrahedron (m²)
\( V \) — Volume of Deltoidal Icositetrahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the surface area from the volume measurement using the specific geometric properties of the deltoidal icositetrahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, materials science, and engineering. It helps determine material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the volume of the deltoidal icositetrahedron in cubic meters. The value must be positive and valid. The calculator will compute the corresponding total surface area.

5. Frequently Asked Questions (FAQ)

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

Q2: What are the applications of this calculation?
A: This calculation is used in crystallography, architectural design, and mathematical modeling of complex geometric structures.

Q3: How accurate is this formula?
A: The formula is mathematically exact for perfect deltoidal icositetrahedron shapes and provides precise surface area calculations.

Q4: Can this calculator handle different units?
A: The calculator uses cubic meters for volume and square meters for surface area. Convert other units to these standards before calculation.

Q5: What if I get an error in calculation?
A: Ensure the volume value is positive and within reasonable limits. Very large or very small values may require scientific notation.