Volume Of Triakis Icosahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left( \frac{12 \times \sqrt{109 - 30\sqrt{5}}}{(5 + 7\sqrt{5}) \times \frac{A}{V}} \right)^3 \]

Volume of Triakis Icosahedron:

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1. What is Volume of Triakis Icosahedron?

The Volume of Triakis Icosahedron is the quantity of three dimensional space enclosed by the entire surface of Triakis Icosahedron. It is a Catalan solid that can be derived from the regular icosahedron by adding pyramids to each face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left( \frac{12 \times \sqrt{109 - 30\sqrt{5}}}{(5 + 7\sqrt{5}) \times \frac{A}{V}} \right)^3 \]

Where:

\( V \) — Volume of Triakis Icosahedron (m³)
\( \frac{A}{V} \) — Surface to Volume Ratio of Triakis Icosahedron (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume of a Triakis Icosahedron based on its surface to volume ratio, using mathematical constants derived from the geometry of the shape.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in mathematics, engineering, and various scientific fields. For the Triakis Icosahedron, volume calculation helps in understanding its spatial properties and applications in crystallography, architecture, and molecular modeling.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in m⁻¹. The value must be positive and greater than zero. The calculator will compute the corresponding volume of the Triakis Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that can be obtained by adding a triangular pyramid to each face of a regular icosahedron, resulting in a polyhedron with 60 faces.

Q2: What are the units for surface to volume ratio?
A: Surface to volume ratio is typically measured in inverse meters (m⁻¹), representing the surface area per unit volume.

Q3: Can this calculator handle different units?
A: The calculator expects input in meters and meters⁻¹. For other units, appropriate conversion should be applied before input.

Q4: What is the significance of the mathematical constants in the formula?
A: The constants (5, 7, 44, 12, 109, 30) are derived from the geometric properties of the Triakis Icosahedron and the golden ratio relationships inherent in icosahedral symmetry.

Q5: What are practical applications of Triakis Icosahedron volume calculations?
A: These calculations are used in crystallography for certain crystal structures, in architectural design for geodesic domes, and in molecular modeling for fullerene and virus structures.