Insphere Radius of Pentagonal Icositetrahedron given Long Edge Calculator

Formula Used:

\[ r_i = \frac{l_{e(Long)}}{\sqrt{(2-C) \times (3-C) \times (C+1)}} \]

Insphere Radius of Pentagonal Icositetrahedron:

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1. What is the Insphere Radius of Pentagonal Icositetrahedron?

The Insphere Radius of Pentagonal Icositetrahedron is the radius of the sphere that the Pentagonal Icositetrahedron contains in such a way that all the faces touch the sphere. It represents the largest sphere that can fit inside the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{l_{e(Long)}}{\sqrt{(2-C) \times (3-C) \times (C+1)}} \]

Where:

\( r_i \) — Insphere Radius of Pentagonal Icositetrahedron (m)
\( l_{e(Long)} \) — Long Edge of Pentagonal Icositetrahedron (m)
\( C \) — Tribonacci constant (≈1.839286755214161)

Explanation: The formula calculates the insphere radius based on the long edge length of the pentagonal icositetrahedron, using the mathematical constant known as the Tribonacci constant.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and crystallography for understanding the spatial properties of polyhedra. It helps determine the maximum size of a sphere that can be inscribed within the polyhedron while touching all its faces.

4. Using the Calculator

Tips: Enter the long edge length of the pentagonal icositetrahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A Pentagonal Icositetrahedron is a Catalan solid with 24 pentagonal faces, 38 vertices, and 60 edges. It is 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: How accurate is this calculation?
A: The calculation is mathematically exact when using the precise value of the Tribonacci constant and follows the geometric properties of the pentagonal icositetrahedron.

Q4: 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 relationship with the Tribonacci constant.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical research, crystallography, and the study of polyhedral geometry to understand the spatial relationships within this specific Catalan solid.