Volume Of Truncated Icosidodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = 5 \times \left( \sqrt{ \frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + (2 \times \sqrt{5})})} } \right)^3 \times (19 + (10 \times \sqrt{5})) \]

Volume of Truncated Icosidodecahedron:

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1. What is the Volume of Truncated Icosidodecahedron?

The volume of a truncated icosidodecahedron is the total quantity of three dimensional space enclosed by the surface of this Archimedean solid. It's a complex polyhedron with 120 vertices, 180 edges, and 62 faces.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ V = 5 \times \left( \sqrt{ \frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + (2 \times \sqrt{5})})} } \right)^3 \times (19 + (10 \times \sqrt{5})) \]

Where:

\( V \) — Volume of the truncated icosidodecahedron (m³)
\( TSA \) — Total surface area of the truncated icosidodecahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the volume from the total surface area using the geometric properties and mathematical relationships specific to the truncated icosidodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of complex polyhedra is essential in various fields including architecture, materials science, crystallography, and mathematical research. It helps in understanding spatial properties 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 of the truncated icosidodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosidodecahedron?
A: It's an Archimedean solid obtained by truncating both an icosahedron and a dodecahedron, resulting in a polyhedron with 62 faces (30 squares, 20 hexagons, and 12 decagons).

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the truncated icosidodecahedron, which involves multiple irrational numbers (√3, √5) in its mathematical relationships.

Q3: What are typical applications of this calculation?
A: This calculation is used in architectural design, molecular modeling, geometric art, and mathematical research involving polyhedral structures.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula, assuming precise input values and proper implementation of the mathematical operations.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the truncated icosidodecahedron. Other polyhedra have their own unique volume formulas based on their geometric properties.