Total Surface Area Of Pentagonal Hexecontahedron Given Snub Dodecahedron Edge Calculator

Formula Used:

\[ TSA = 30 \times \left( \frac{le(Snub\ Dodecahedron)}{\sqrt{2+2 \times 0.4715756}} \right)^2 \times (2+3 \times 0.4715756) \times \frac{\sqrt{1-0.4715756^2}}{1-2 \times 0.4715756^2} \]

Total Surface Area of Pentagonal Hexecontahedron:

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1. What is Total Surface Area of Pentagonal Hexecontahedron?

The Total Surface Area of Pentagonal Hexecontahedron is the amount or quantity of two dimensional space covered on the surface of Pentagonal Hexecontahedron. It is a Catalan solid that is dual to the snub dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 30 \times \left( \frac{le(Snub\ Dodecahedron)}{\sqrt{2+2 \times 0.4715756}} \right)^2 \times (2+3 \times 0.4715756) \times \frac{\sqrt{1-0.4715756^2}}{1-2 \times 0.4715756^2} \]

Where:

\( TSA \) — Total Surface Area of Pentagonal Hexecontahedron (m²)
\( le(Snub\ Dodecahedron) \) — Length of any edge of the Snub Dodecahedron (m)
0.4715756 — A constant value used in the calculation

Explanation: This formula calculates the surface area based on the edge length of the dual snub dodecahedron, using specific mathematical constants and operations.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is important in various fields including mathematics, engineering, architecture, and material science. It helps in determining material requirements, heat transfer properties, and other physical characteristics.

4. Using the Calculator

Tips: Enter the snub dodecahedron edge length in meters. The value must be positive and valid. The calculator will compute the total surface area of the pentagonal hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a pentagonal hexecontahedron?
A: A pentagonal hexecontahedron is a Catalan solid with 60 faces, each of which is an irregular pentagon. It is the dual of the snub dodecahedron.

Q2: Why is the constant 0.4715756 used in the formula?
A: This constant is derived from the mathematical properties of the pentagonal hexecontahedron and its relationship with the snub dodecahedron.

Q3: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for calculating the surface area of a pentagonal hexecontahedron given the edge length of its dual snub dodecahedron.

Q4: What units does the calculator use?
A: The calculator uses meters for input and square meters for output. Ensure consistent units for accurate results.

Q5: How accurate is the calculation?
A: The calculation is mathematically precise based on the formula, though practical accuracy depends on the precision of the input value.