Surface to Volume Ratio of Pentagonal Icositetrahedron given Insphere Radius Calculator

Formula Used:

\[ SA:V = \frac{3\sqrt{\frac{22(5[Tribonacci_C]-1)}{(4[Tribonacci_C])-3}}}{(2\sqrt{(2-[Tribonacci_C])(3-[Tribonacci_C])} \times r_i) \times \sqrt{\frac{11([Tribonacci_C]-4)}{2((20[Tribonacci_C])-37)}}} \]

Surface to Volume Ratio (SA:V):

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1. What is the Surface to Volume Ratio of Pentagonal Icositetrahedron?

The Surface to Volume Ratio (SA:V) of a Pentagonal Icositetrahedron is a geometric property that relates the total surface area to the total volume of this polyhedron. It's an important parameter in materials science, chemistry, and physics where surface properties relative to volume are significant.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( r_i \) — Insphere radius (m)

Explanation: This complex formula accounts for the unique geometry of the pentagonal icositetrahedron and its relationship between surface area, volume, and the insphere radius.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in many scientific applications including catalysis, heat transfer, biological systems, and materials science where surface interactions dominate bulk properties.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-zero. The calculator will compute the surface to volume ratio using the specialized formula for pentagonal icositetrahedrons.

5. Frequently Asked Questions (FAQ)

Q1: What is a pentagonal icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 faces, each of which is an irregular pentagon. It is the dual of the snub cube.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the ratio toward which adjacent Tribonacci numbers tend, approximately equal to 1.839286755214161.

Q3: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can be contained within the polyhedron, touching all faces.

Q4: Why is surface to volume ratio important?
A: It's critical in processes where surface interactions are important, such as chemical reactions, heat transfer, and biological processes.

Q5: What are typical values for this ratio?
A: The ratio depends on the insphere radius, but generally decreases as the size of the polyhedron increases.