Total Surface Area of Pentagonal Icositetrahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = 3 \times (2 \times \sqrt{(2-[Tribonacci_C]) \times (3-[Tribonacci_C])} \times r_i)^2 \times \sqrt{\frac{22 \times (5 \times [Tribonacci_C]-1)}{(4 \times [Tribonacci_C])-3}} \]

Total Surface Area of Pentagonal Icositetrahedron:

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

The Pentagonal Icositetrahedron is a Catalan solid with 24 congruent irregular pentagonal faces. Its total surface area represents the sum of the areas of all these faces, providing a measure of the complete exterior surface of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ TSA = 3 \times (2 \times \sqrt{(2-[Tribonacci_C]) \times (3-[Tribonacci_C])} \times r_i)^2 \times \sqrt{\frac{22 \times (5 \times [Tribonacci_C]-1)}{(4 \times [Tribonacci_C])-3}} \]

Where:

\( TSA \) — Total Surface Area (m²)
\( r_i \) — Insphere Radius (m)
\( [Tribonacci_C] \) — Tribonacci constant ≈ 1.839286755214161

Explanation: This formula relates the total surface area to the insphere radius through the mathematical constant [Tribonacci_C], which is characteristic of this specific polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of polyhedra is essential in geometry, materials science, and engineering applications where surface properties affect material behavior, heat transfer, or other physical characteristics.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-zero. The calculator will compute the total surface area using the specialized formula for the Pentagonal Icositetrahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: It's a Catalan solid with 24 faces, each being an irregular pentagon. It's the dual of the snub cube and has interesting geometric properties.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161.

Q3: How is the insphere radius defined?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, tangent to all its faces.

Q4: What are practical applications of this calculation?
A: This calculation is useful in crystallography, materials design, and architectural geometry where this specific polyhedral form appears.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Pentagonal Icositetrahedron due to its unique geometric properties and the involvement of the Tribonacci constant.