Total Surface Area of Truncated Dodecahedron given Volume Calculator

Formula Used:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{12 \times V}{5 \times (99 + 47 \times \sqrt{5})} \right)^{\frac{2}{3}} \]

Total Surface Area of Truncated Dodecahedron:

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

The Total Surface Area of a Truncated Dodecahedron is the total quantity of plane enclosed by the entire surface of this polyhedron. It's calculated based on the volume of the solid using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{12 \times V}{5 \times (99 + 47 \times \sqrt{5})} \right)^{\frac{2}{3}} \]

Where:

\( TSA \) — Total Surface Area (m²)
\( V \) — Volume of Truncated Dodecahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the surface area from the volume by establishing the relationship between these two geometric properties of the truncated dodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, heat transfer calculations, and understanding the geometric properties of this complex polyhedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the volume of the truncated dodecahedron in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

Q2: Why is the formula so complex?
A: The complexity arises from the mathematical relationship between volume and surface area in this specific polyhedron, involving irrational numbers and fractional exponents.

Q3: What are typical volume values for practical applications?
A: Volume values can vary widely depending on the scale, from small mathematical models to large architectural structures, typically ranging from cubic centimeters to cubic meters.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric form. In practical applications, manufacturing tolerances and material properties may affect the actual surface area.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated dodecahedra. Other polyhedra have different mathematical relationships between volume and surface area.