Surface To Volume Ratio Of Truncated Icosahedron Given Volume Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{\left(\frac{4V}{125 + 43\sqrt{5}}\right)^{1/3} \times (125 + 43\sqrt{5})} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Truncated Icosahedron?

The surface to volume ratio of a truncated icosahedron is a geometric property that represents the relationship between the total surface area and the volume of this polyhedron. A truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons), 90 edges, and 60 vertices.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ \text{Surface to Volume Ratio} = \frac{12 \times (10\sqrt{3} + \sqrt{25 + 10\sqrt{5}})}{\left(\frac{4V}{125 + 43\sqrt{5}}\right)^{1/3} \times (125 + 43\sqrt{5})} \]

Where:

\( V \) — Volume of the truncated icosahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the ratio of surface area to volume based on the given volume of the truncated icosahedron, using the mathematical constants and relationships specific to this geometric shape.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is an important geometric property that influences various physical and chemical properties of materials and structures. In materials science, engineering, and physics, this ratio affects properties such as heat transfer, chemical reactivity, strength-to-weight ratio, and other surface-dependent phenomena.

4. Using the Calculator

Tips: Enter the volume of the truncated icosahedron in cubic meters. The volume must be a positive value greater than zero. The calculator will compute the corresponding surface to volume ratio.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid obtained by truncating the vertices of a regular icosahedron. It has 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

Q2: What are real-world applications of truncated icosahedrons?
A: The most famous example is the soccer ball, which has the shape of a truncated icosahedron. They are also used in architecture, molecular structures (fullerenes), and various engineering applications.

Q3: Why is surface to volume ratio important?
A: This ratio is crucial in many fields as it affects how objects interact with their environment. Higher ratios mean more surface area relative to volume, which can enhance processes like heat dissipation, chemical reactions, and mass transfer.

Q4: How does the ratio change with size?
A: For similar shapes, as size increases, the surface to volume ratio decreases. This is why smaller objects have relatively more surface area compared to their volume.

Q5: Can this calculator be used for other polyhedrons?
A: No, this specific calculator is designed only for truncated icosahedrons. Other polyhedrons have different mathematical formulas for calculating their surface to volume ratios.