Total Surface Area of Triakis Icosahedron given Volume Calculator

Formula Used:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{44V}{5(5 + 7\sqrt{5})} \right)^{\frac{2}{3}} \]

Total Surface Area of Triakis Icosahedron:

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

The Total Surface Area of Triakis Icosahedron is the amount or quantity of two dimensional space covered on the surface of Triakis Icosahedron. It's an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{44V}{5(5 + 7\sqrt{5})} \right)^{\frac{2}{3}} \]

Where:

\( TSA \) — Total Surface Area of Triakis Icosahedron (m²)
\( V \) — Volume of Triakis Icosahedron (m³)
\( \sqrt{} \) — Square root function

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

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric shapes is crucial in various fields including architecture, engineering, material science, and 3D modeling. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the volume of the Triakis Icosahedron in cubic meters. The volume must be a positive value 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. It has 60 faces, 90 edges, and 32 vertices.

Q2: Why is this formula so complex?
A: The formula incorporates mathematical constants and relationships specific to the geometry of the Triakis Icosahedron, making it more complex than formulas for simpler shapes.

Q3: What units should I use?
A: Use consistent units throughout. If volume is in cubic meters, the surface area result will be in square meters.

Q4: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this formula accurate for all Triakis Icosahedrons?
A: Yes, this formula is mathematically derived and provides accurate results for any regular Triakis Icosahedron when given its volume.