Total Surface Area Of Great Stellated Dodecahedron Given Pentagram Chord Calculator

Formula Used:

\[ TSA = 15 \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{lc(Pentagram)}{2 + \sqrt{5}} \right)^2 \]

Total Surface Area of Great Stellated Dodecahedron:

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

The Total Surface Area of a Great Stellated Dodecahedron is the total area of all its faces. It is a complex polyhedron with pentagrammic faces, and its surface area calculation involves specific geometric relationships.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{lc(Pentagram)}{2 + \sqrt{5}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Great Stellated Dodecahedron (m²)
\( lc(Pentagram) \) — Pentagram Chord of Great Stellated Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area based on the pentagram chord length, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra is important in geometry, architectural design, and materials science where understanding surface properties is crucial.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It is one of the Kepler-Poinsot polyhedra, formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron.

Q2: What is the Pentagram Chord in this context?
A: The pentagram chord is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Great Stellated Dodecahedron.

Q3: What are the units for surface area?
A: The surface area is calculated in square meters (m²), but can be converted to other area units as needed.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric form and may not account for manufacturing tolerances or material properties in real-world applications.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is designed only for the Great Stellated Dodecahedron and its unique geometric properties.