Surface To Volume Ratio Of Pentagonal Icositetrahedron Given Midsphere Radius Calculator

Formula Used:

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

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 is a geometric property that represents the relationship between the total surface area and the total volume of this polyhedron. It's an important parameter in materials science and geometry.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

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

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( r_m \) — Midsphere radius of the Pentagonal Icositetrahedron (m)

Explanation: This complex formula incorporates the Tribonacci constant and accounts for the unique geometric properties of the Pentagonal Icositetrahedron.

3. Importance of Surface to Volume Ratio

Details: The SA:V ratio is crucial in various applications including material science, chemical reactivity studies, heat transfer analysis, and biological systems where surface area to volume relationships affect physical properties.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and non-zero. The calculator automatically uses the Tribonacci constant in the computation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A Pentagonal Icositetrahedron is a Catalan solid with 24 faces, each being an irregular pentagon. It's 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: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q4: What are typical SA:V values for this shape?
A: The SA:V ratio depends on the size of the polyhedron. Smaller polyhedra have higher SA:V ratios, while larger ones have lower ratios.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived specifically for the Pentagonal Icositetrahedron due to its unique geometric properties.