Total Surface Area Of Great Dodecahedron Calculator

Formula Used:

\[ TSA = 15 \times \sqrt{5 - (2 \times \sqrt{5})} \times le^2 \]

Total Surface Area (TSA):

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

The Total Surface Area of a Great Dodecahedron is the total quantity of plane enclosed on the entire surface of this polyhedron. It is a star polyhedron with pentagrammic faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 - (2 \times \sqrt{5})} \times le^2 \]

Where:

\( TSA \) — Total Surface Area (m²)
\( le \) — Edge Length (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area based on the edge length of the Great Dodecahedron, incorporating the mathematical constant and geometric properties specific to this shape.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is important in geometry, architecture, and material science for understanding the spatial properties and material requirements of this complex polyhedral shape.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagrammic faces intersecting each other.

Q2: How is this different from a regular dodecahedron?
A: While both have 12 faces, a regular dodecahedron has pentagonal faces, while a Great Dodecahedron has pentagrammic (star-shaped) faces that intersect.

Q3: What are the units for the result?
A: The result is in square meters (m²), consistent with the input edge length unit.

Q4: Can this formula be used for any polyhedron?
A: No, this specific formula applies only to the Great Dodecahedron due to its unique geometric properties.

Q5: What is the significance of the constant in the formula?
A: The constant \( \sqrt{5 - (2 \times \sqrt{5})} \) is derived from the geometric properties and golden ratio relationships inherent in the Great Dodecahedron's structure.