Insphere Radius of Pentagonal Icositetrahedron given Short Edge Calculator

Formula Used:

\[ r_i = \frac{1}{2} \times \sqrt{\frac{[Tribonacci_C] + 1}{(2 - [Tribonacci_C]) \times (3 - [Tribonacci_C])}} \times l_e(Short) \]

Insphere Radius of Pentagonal Icositetrahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 touching all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{1}{2} \times \sqrt{\frac{[Tribonacci_C] + 1}{(2 - [Tribonacci_C]) \times (3 - [Tribonacci_C])}} \times l_e(Short) \]

Where:

\( r_i \) — Insphere Radius of Pentagonal Icositetrahedron (m)
\( [Tribonacci_C] \) — Tribonacci constant (≈ 1.839286755214161)
\( l_e(Short) \) — Short Edge of Pentagonal Icositetrahedron (m)

Explanation: This formula calculates the insphere radius based on the short edge length of the pentagonal icositetrahedron, using the mathematical constant Tribonacci_C which is specific to this polyhedral structure.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and crystallography for understanding the spatial properties of polyhedra, determining packing efficiency, and analyzing the geometric relationships between different elements of the structure.

4. Using the Calculator

Tips: Enter the short edge length of the pentagonal icositetrahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A Pentagonal Icositetrahedron is a Catalan solid with 24 pentagonal faces. It is the dual polyhedron 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 for a perfect pentagonal icositetrahedron, using the precise Tribonacci constant value.

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 the Tribonacci constant relationship.

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, materials science, and geometric modeling where pentagonal icositetrahedron structures appear.