1. What is the Circumsphere Radius of Truncated Icosahedron?

The circumsphere radius of a truncated icosahedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the sphere's surface. The truncated icosahedron is an Archimedean solid with 32 faces (12 pentagons and 20 hexagons).

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumsphere radius (m)
• \( TSA \) — Total surface area of the truncated icosahedron (m²)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
• \( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula relates the circumsphere radius to the total surface area through geometric relationships specific to the truncated icosahedron's structure.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, materials science, and architecture for understanding the spatial properties and packing efficiency of truncated icosahedral structures.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

Q2: Where is the truncated icosahedron found in nature?
A: The truncated icosahedron is best known as the shape of a soccer ball and is also found in certain carbon molecules (fullerenes).

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the size of the polyhedron. For a standard soccer ball, it would be approximately half the ball's diameter.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated icosahedra. Other polyhedra have different geometric relationships.

Q5: What units should I use?
A: The calculator uses meters for length and square meters for area, but you can use any consistent unit system as long as you maintain consistency.