Total Surface Area of Truncated Icosahedron given Circumsphere Radius Calculator

Formula Used:

\[ TSA = 3 \times \left( \frac{4 \times r_c}{\sqrt{58 + (18 \times \sqrt{5})}} \right)^2 \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

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 polyhedron. A truncated icosahedron is an Archimedean solid with 32 faces - 12 regular pentagons and 20 regular hexagons.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 3 \times \left( \frac{4 \times r_c}{\sqrt{58 + (18 \times \sqrt{5})}} \right)^2 \times \left( (10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})} \right) \]

Where:

\( TSA \) — Total Surface Area of Truncated Icosahedron (m²)
\( r_c \) — Circumsphere Radius of Truncated Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the circumsphere radius, incorporating the geometric properties of the truncated icosahedron.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is important in various applications including material science, architecture, and geometric modeling. It helps in determining material requirements, structural properties, and understanding the geometric characteristics of this complex polyhedron.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the total surface area based on the provided radius.

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, 90 edges, and 60 vertices.

Q2: What are the practical applications of this calculation?
A: This calculation is used in various fields including architecture, material science, chemical modeling (fullerenes), and sports equipment design (soccer balls).

Q3: How accurate is this formula?
A: The formula is mathematically exact for a perfect truncated icosahedron and provides precise calculations when accurate inputs are provided.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, or convert the result from square meters to your desired area unit.

Q5: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the truncated icosahedron such that all vertices lie on the sphere's surface.