Total Surface Area of Truncated Dodecahedron given Dodecahedral Edge Length Calculator

Formula Used:

\[ TSA = (\sqrt{3} + (6 \times \sqrt{5 + (2 \times \sqrt{5})})) \times l_{e(Dodecahedron)}^2 \]

Total Surface Area (TSA):

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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 geometric solid. It represents the sum of all its polygonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = (\sqrt{3} + (6 \times \sqrt{5 + (2 \times \sqrt{5})})) \times l_{e(Dodecahedron)}^2 \]

Where:

\( TSA \) — Total Surface Area (m²)
\( l_{e(Dodecahedron)} \) — Dodecahedral Edge Length (m)
\( \sqrt{3} \) — Square root of 3
\( \sqrt{5} \) — Square root of 5

Explanation: The formula combines geometric constants and the square of the dodecahedral edge length to calculate the total surface area.

3. Importance of TSA Calculation

Details: Calculating the total surface area is important in geometry, material science, and engineering applications where surface properties need to be determined.

4. Using the Calculator

Tips: Enter the dodecahedral edge length in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by cutting the corners of a regular dodecahedron.

Q2: What units should be used for input?
A: The calculator uses meters as the default unit for length measurements.

Q3: How accurate is the calculation?
A: The calculation uses precise mathematical constants and provides results with 6 decimal places accuracy.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated dodecahedrons.

Q5: What are the practical applications of this calculation?
A: This calculation is useful in architecture, 3D modeling, material science, and geometric analysis.