Insphere Radius of Pentagonal Icositetrahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{1}{2\sqrt{(2-[Tribonacci_C])(3-[Tribonacci_C])}} \times V^{\frac{1}{3}} \times \left( \frac{2(20[Tribonacci_C]-37)}{11([Tribonacci_C]-4)} \right)^{\frac{1}{6}} \]

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.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{1}{2\sqrt{(2-[Tribonacci_C])(3-[Tribonacci_C])}} \times V^{\frac{1}{3}} \times \left( \frac{2(20[Tribonacci_C]-37)}{11([Tribonacci_C]-4)} \right)^{\frac{1}{6}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( V \) — Volume of Pentagonal Icositetrahedron (m³)
\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)

Explanation: The formula calculates the insphere radius based on the volume of the pentagonal icositetrahedron, using the mathematical constant Tribonacci_C.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and crystallography for understanding the spatial properties and packing efficiency of pentagonal icositetrahedrons.

4. Using the Calculator

Tips: Enter the volume of the pentagonal icositetrahedron in cubic 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 polyhedron with 24 pentagonal faces, 38 vertices, and 60 edges.

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 this formula derived?
A: The formula is derived from geometric relationships and mathematical properties specific to pentagonal icositetrahedrons.

Q4: What are typical values for insphere radius?
A: The insphere radius depends on the volume, but typically ranges from a few centimeters to several meters for practical applications.

Q5: Can this calculator be used for other polyhedrons?
A: No, this specific formula applies only to pentagonal icositetrahedrons due to their unique geometric properties.