Total Surface Area of Truncated Icosahedron given Midsphere Radius Calculator

Formula Used:

\[ TSA = 3 \times \left( \frac{4 \times r_m}{3 \times (1 + \sqrt{5})} \right)^2 \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

Total Surface Area of Truncated Icosahedron:

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

The Total Surface Area of a Truncated Icosahedron is the total quantity of plane enclosed by the entire surface of this geometric solid. A truncated icosahedron is an Archimedean solid with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 3 \times \left( \frac{4 \times r_m}{3 \times (1 + \sqrt{5})} \right)^2 \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

Where:

\( TSA \) — Total Surface Area of Truncated Icosahedron (m²)
\( r_m \) — Midsphere Radius of Truncated Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the total surface area based on the midsphere radius, which is the radius of the sphere tangent to all edges of the truncated icosahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, materials science, chemistry (for surface reactions), and engineering design where surface properties affect functionality and performance.

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 using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid obtained by truncating the vertices of a regular icosahedron. It has 32 faces (12 pentagons and 20 hexagons) and is best known as the shape of a soccer ball.

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. For a truncated icosahedron, it's the sphere that touches every edge at exactly one point.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived specifically for the truncated icosahedron. Other polyhedra have different surface area formulas based on their unique geometric properties.

Q4: What are practical applications of this calculation?
A: This calculation is used in materials science, nanotechnology (fullerene molecules), architectural design, and anywhere the surface properties of this specific geometric shape need to be quantified.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy of the result depends on the precision of the input value and the computational precision of the calculator.