Total Surface Area Of Triakis Icosahedron Given Midsphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{4 \times r_m}{1 + \sqrt{5}} \right)^2 \]

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 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:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{4 \times r_m}{1 + \sqrt{5}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Triakis Icosahedron (m²)
\( r_m \) — Midsphere Radius of Triakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the total surface area based on the midsphere radius, incorporating the mathematical constant √5 which is fundamental to icosahedral geometry.

3. Importance of Surface Area Calculation

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

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the total surface area in square meters.

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 is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is only applicable to the Triakis Icosahedron geometry.

Q4: What are practical applications of this calculation?
A: This calculation is useful in crystallography, molecular modeling, architectural design, and any field dealing with complex geometric structures.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using the precise formula, though computational rounding may introduce minor errors in decimal representation.