Total Surface Area of Pentagonal Icositetrahedron given Midsphere Radius Calculator

Formula Used:

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

Total Surface Area (TSA):

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

The Pentagonal Icositetrahedron is a Catalan solid with 24 pentagonal faces. Its total surface area represents the sum of the areas of all its faces, providing important geometric information about this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

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

Where:

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

Explanation: This formula relates the total surface area to the midsphere radius through the mathematical constant known as the Tribonacci constant, which appears in various geometric contexts.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra like the Pentagonal Icositetrahedron is crucial for material science applications, architectural design, and understanding geometric properties of three-dimensional shapes.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the total surface area using the specialized formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: It's a Catalan solid with 24 pentagonal faces, 60 edges, and 38 vertices. It's the dual of the snub cube.

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: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q4: Are there other ways to calculate the surface area?
A: Yes, the surface area can also be calculated from edge length or other geometric parameters, but the midsphere radius provides a unique approach.

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, and mathematical research involving polyhedral geometry.