Total Surface Area of Small Stellated Dodecahedron given Pentagram Chord Calculator

Formula Used:

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

Total Surface Area of Small Stellated Dodecahedron:

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

The Total Surface Area of a Small Stellated Dodecahedron is the total area of all its faces. It's a complex polyhedron with pentagrammic faces that extend outward from a central dodecahedron, creating a star-like shape.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area of Small Stellated Dodecahedron
\( l_{c(Pentagram)} \) — Pentagram Chord of Small Stellated Dodecahedron
\( \sqrt{5 + 2\sqrt{5}} \) — Mathematical constant related to the geometry of pentagons
\( 2 + \sqrt{5} \) — Scaling factor derived from the golden ratio

Explanation: The formula calculates the surface area based on the pentagram chord length, incorporating geometric constants specific to the pentagonal symmetry of the shape.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is important in geometry, materials science, and architectural design where this complex shape might be used. It helps in determining material requirements and understanding the geometric properties of this unique polyhedron.

4. Using the Calculator

Tips: Enter the pentagram chord length in meters. The value must be positive and greater than zero. The calculator will compute the total surface area based on the geometric relationship between the pentagram chord and the surface area.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's one of the Kepler-Poinsot polyhedra, created by extending the faces of a regular dodecahedron until they meet to form pentagram-shaped faces.

Q2: How is the pentagram chord defined?
A: The pentagram chord is the distance between any pair of non-adjacent peak vertices of the pentagram faces of the Small Stellated Dodecahedron.

Q3: What are the applications of this calculation?
A: This calculation is used in mathematical research, architectural design, and in understanding the geometric properties of complex polyhedra.

Q4: Are there other ways to calculate the surface area?
A: Yes, the surface area can also be calculated using the edge length or other geometric parameters, but the pentagram chord provides a direct relationship.

Q5: What is the precision of this calculation?
A: The calculation uses mathematical constants with high precision, but the result's accuracy depends on the precision of the input measurement.