Total Surface Area of Triakis Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{12 \times \sqrt{109 - 30\sqrt{5}}}{(5 + 7\sqrt{5}) \times \frac{S}{V}} \right)^2 \]

Total Surface Area (TSA):

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1. What is the Total Surface Area of Triakis Icosahedron?

The Total Surface Area of a Triakis Icosahedron is the total area of all its faces. A Triakis Icosahedron is a polyhedron created by attaching a triangular pyramid to each face of a regular icosahedron, resulting in 60 isosceles triangular faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{12 \times \sqrt{109 - 30\sqrt{5}}}{(5 + 7\sqrt{5}) \times \frac{S}{V}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area (m²)
\( \frac{S}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the surface to volume ratio of the Triakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is important in various fields including materials science, chemistry (for surface reactions), and engineering (for heat transfer and fluid dynamics applications involving polyhedral structures).

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. 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 created by attaching a triangular pyramid to each face of a regular icosahedron, resulting in 60 faces.

Q2: What are the applications of this calculation?
A: This calculation is used in crystallography, nanotechnology, architectural design, and any field dealing with polyhedral structures and their surface properties.

Q3: How accurate is this formula?
A: The formula is mathematically exact for perfect Triakis Icosahedron shapes and provides precise calculations based on geometric principles.

Q4: Can this calculator handle different units?
A: The calculator uses meters for length units. For other units, appropriate conversion factors must be applied to the input values before calculation.

Q5: What is the typical range of surface to volume ratios?
A: The surface to volume ratio depends on the size of the polyhedron. Smaller structures typically have higher surface to volume ratios.