Surface to Volume Ratio of Pentagonal Icositetrahedron given Short Edge Calculator

Formula Used:

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

Surface to Volume Ratio (SA:V):

Unit Converter ▲

Unit Converter ▼

From:	To:

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

The Surface to Volume Ratio (SA:V) of a Pentagonal Icositetrahedron is a geometric measurement that compares the total surface area to the total volume of this polyhedron. It's an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

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

Where:

\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)
\( l_e(Short) \) — Short edge length of the pentagonal icositetrahedron

Explanation: This complex formula accounts for the unique geometric properties of the pentagonal icositetrahedron and its relationship with the Tribonacci constant.

3. Importance of SA:V Calculation

Details: The surface to volume ratio is crucial in materials science, heat transfer studies, and geometric analysis where the relationship between surface area and volume affects physical properties and behavior.

4. Using the Calculator

Tips: Enter the short edge length of the pentagonal icositetrahedron in meters. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 pentagonal faces, 60 edges, and 38 vertices.

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 are typical SA:V values for this shape?
A: The SA:V ratio varies inversely with the edge length. Smaller edge lengths yield higher SA:V ratios, while larger edge lengths yield lower ratios.

Q4: What units are used in this calculation?
A: The SA:V ratio is expressed in m⁻¹ (per meter), as it represents area per unit volume.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the pentagonal icositetrahedron and incorporates the unique mathematical properties of this particular shape.