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Insphere Radius of Pentagonal Icositetrahedron Given Total Surface Area Calculator

Formula:

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

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1. What is the Insphere Radius of Pentagonal Icositetrahedron?

The insphere radius of a Pentagonal Icositetrahedron is the radius of the largest sphere that can be contained within the polyhedron such that it touches all its faces. This measurement is important in understanding the geometric properties of this Catalan solid.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

Explanation: This formula relates the insphere radius to the total surface area using the mathematical constant specific to this polyhedron's geometry.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius helps in understanding the internal geometry of the Pentagonal Icositetrahedron, which has applications in crystallography, materials science, and mathematical research on Catalan solids.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and non-zero. The calculator uses the fixed Tribonacci constant value of approximately 1.839286755214161.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: It's a Catalan solid with 24 identical 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: 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.

Q4: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a Pentagonal Icositetrahedron with TSA of 1m², the insphere radius is approximately 0.095m.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect Pentagonal Icositetrahedron, assuming precise measurement of the total surface area.

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