Surface To Volume Ratio Of Great Icosahedron Given Short Ridge Length Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{3\sqrt{3}(5+4\sqrt{5})}{\frac{1}{4}(25+9\sqrt{5})} \cdot \frac{\sqrt{10}}{5 \cdot \text{Short Ridge Length}} \]

Surface to Volume Ratio:

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

The Surface to Volume Ratio of a Great Icosahedron is a geometric measurement that compares the total surface area to the volume of this complex polyhedron. It provides insight into the spatial efficiency and geometric properties of the shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{3\sqrt{3}(5+4\sqrt{5})}{\frac{1}{4}(25+9\sqrt{5})} \cdot \frac{\sqrt{10}}{5 \cdot \text{Short Ridge Length}} \]

Where:

Short Ridge Length — The length of the short ridge of the Great Icosahedron (in meters)
\(\sqrt{}\) — Square root function

Explanation: This formula calculates the ratio of surface area to volume based on the geometric properties of the Great Icosahedron and its short ridge length.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and physics. For geometric shapes like the Great Icosahedron, this ratio helps understand the relationship between the external surface and internal volume of the shape.

4. Using the Calculator

Tips: Enter the short ridge length of the Great Icosahedron in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: A Great Icosahedron is a non-convex regular polyhedron with 20 triangular faces. It is one of the Kepler-Poinsot polyhedra.

Q2: Why is surface to volume ratio important?
A: Surface to volume ratio affects many physical and chemical properties including heat transfer, reaction rates, and structural strength.

Q3: What units are used in this calculation?
A: The input is in meters and the result is in reciprocal meters (m⁻¹), which is the standard unit for surface to volume ratio.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Great Icosahedron shape with its unique geometric properties.

Q5: What if I have the long ridge length instead?
A: This calculator specifically requires the short ridge length. For calculations with long ridge length, a different formula would be needed.