Total Surface Area of Truncated Icosahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = 3 \times \left( \frac{12 \times ((10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})})}{RA/V \times (125 + (43 \times \sqrt{5}))} \right)^2 \times ((10 \times \sqrt{3}) + \sqrt{25 + (10 \times \sqrt{5})}) \]

Total Surface Area (TSA):

Unit Converter ▲

Unit Converter ▼

From:	To:

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.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( TSA \) — Total Surface Area (m²)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula derives the total surface area from the surface to volume ratio using the geometric properties of the truncated icosahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material science, architecture, and engineering where surface properties affect functionality, coating requirements, and structural integrity.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid formed by truncating the vertices of a regular icosahedron, resulting in 32 faces (12 pentagons and 20 hexagons).

Q2: What are real-world applications of truncated icosahedrons?
A: The most famous example is the soccer ball pattern. They're also used in molecular structures (fullerenes) and architectural designs.

Q3: How is surface to volume ratio related to surface area?
A: Surface to volume ratio (RA/V) = Total Surface Area / Volume. This calculator uses this relationship to derive surface area from the given ratio.

Q4: What units should I use for input and output?
A: Input surface to volume ratio in 1/meter, and the output total surface area will be in square meters (m²).

Q5: Are there limitations to this calculation?
A: This formula assumes a perfect geometric truncated icosahedron. Real-world objects may have manufacturing tolerances that affect the actual surface area.