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Total Surface Area of Great Stellated Dodecahedron given Circumradius Calculator

Formula Used:

\[ TSA = 15 \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (3 + \sqrt{5})} \right)^2 \]

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

The Great Stellated Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagram faces. Its total surface area represents the sum of the areas of all its faces, providing a measure of the polyhedron's external coverage.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (3 + \sqrt{5})} \right)^2 \]

Where:

Explanation: This formula derives the surface area from the circumradius, incorporating geometric constants specific to the Great Stellated Dodecahedron's structure.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is essential in geometry for understanding the polyhedron's properties, and in applied contexts like material science for coating or painting estimations.

4. Using the Calculator

Tips: Enter the circumradius in meters. The value must be positive. The result will be the total surface area in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It is a regular star polyhedron with 12 pentagram faces, 30 edges, and 20 vertices, and is one of the four Kepler-Poinsot solids.

Q2: How is circumradius defined for this polyhedron?
A: The circumradius is the radius of a sphere that passes through all the vertices of the Great Stellated Dodecahedron.

Q3: Can the formula be used for other units?
A: Yes, as long as the circumradius and resulting surface area use consistent units (e.g., cm for circumradius gives cm² for area).

Q4: What are typical values for the circumradius?
A: The circumradius depends on the size of the polyhedron. There is no typical value; it varies with the specific instance.

Q5: Is the Great Stellated Dodecahedron found in nature?
A: While not common, similar geometric patterns can appear in crystal structures and certain natural formations.

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