Surface to Volume Ratio of Pentagonal Icositetrahedron Calculator

Formula Used:

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

Surface to Volume Ratio (SA:V):

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

The Surface to Volume Ratio (SA:V) of a Pentagonal Icositetrahedron represents the relationship between its total surface area and its total volume. It's an important geometric property that indicates how much surface area is available per unit volume of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( l_e \) — Snub Cube Edge length (m)

Explanation: This formula calculates the surface to volume ratio based on the geometric properties of the Pentagonal Icositetrahedron and its relationship to the Snub Cube through duality.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. For polyhedra, it helps understand properties like heat transfer, diffusion rates, and structural efficiency.

4. Using the Calculator

Tips: Enter the Snub Cube Edge length in meters. The value must be positive and greater than 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: A Pentagonal Icositetrahedron is a Catalan solid that is the dual of the Snub Cube. It has 24 faces, each of which is an irregular pentagon.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the ratio toward which adjacent Tribonacci numbers tend. It is the real root of the equation x³ - x² - x - 1 = 0.

Q3: Why is the Snub Cube Edge used in this calculation?
A: The Pentagonal Icositetrahedron is the dual polyhedron of the Snub Cube, meaning their geometric properties are interrelated through duality.

Q4: What are typical values for surface to volume ratio?
A: The value depends on the edge length. For a given shape, smaller objects have higher surface to volume ratios than larger ones of the same shape.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Pentagonal Icositetrahedron. Other polyhedra have different formulas for calculating their surface to volume ratios.