Volume Of Deltoidal Icositetrahedron Given Total Surface Area Calculator

Formula Used:

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

Volume (V):

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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 within this polyhedron, calculated based on its total surface area.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula derives the volume from the total surface area using mathematical constants and geometric relationships 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 3D modeling. It helps in understanding spatial relationships and material requirements.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Icositetrahedron?
A: It's a Catalan solid with 24 congruent deltoid faces, 26 vertices, and 48 edges. It's the dual of the rhombicuboctahedron.

Q2: Why is the formula so complex?
A: The complexity arises from the mathematical relationships between surface area and volume in this specific polyhedron, involving irrational numbers and nested square roots.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, crystallography, and mathematical research involving polyhedra.

Q4: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Deltoidal Icositetrahedron. Other polyhedra have different volume formulas based on their unique geometric properties.

Q5: How accurate is the calculated volume?
A: The calculation is mathematically exact based on the input surface area, though practical measurements may introduce some error in real-world applications.