Surface to Volume Ratio of Triakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{109-30\sqrt{5}}}{5+7\sqrt{5}} \times \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4 \times \text{Insphere Radius}} \]

Surface to Volume Ratio:

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

The Surface to Volume Ratio of a Triakis Icosahedron is a geometric property that represents the relationship between the total surface area and the total volume of this polyhedron. It is an important parameter in materials science and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{109-30\sqrt{5}}}{5+7\sqrt{5}} \times \frac{\sqrt{\frac{10(33+13\sqrt{5})}{61}}}{4 \times \text{Insphere Radius}} \]

Where:

Insphere Radius - The radius of the sphere inscribed within the Triakis Icosahedron (in meters)

Explanation: The formula combines geometric constants and the insphere radius to calculate the surface to volume ratio of this specific polyhedral shape.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in various applications including heat transfer analysis, chemical reactivity studies, and material science where the relationship between surface area and volume affects physical properties and behavior.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron, featuring 60 isosceles triangular faces.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which affects properties like heat dissipation, chemical reactivity, and structural strength.

Q3: What units are used in this calculation?
A: The insphere radius is input in meters, and the result is given in reciprocal meters (m⁻¹).

Q4: Can this calculator handle very small or large values?
A: Yes, but extremely small values may approach computational limits, while very large values should be used with consideration of practical significance.

Q5: Is this formula specific to Triakis Icosahedron?
A: Yes, this formula is specifically derived for the Triakis Icosahedron geometry and may not apply to other polyhedral shapes.