Pentagram Chord Of Small Stellated Dodecahedron Given Total Surface Area Calculator

Formula Used:

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

Pentagram Chord of Small Stellated Dodecahedron:

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1. What is the Pentagram Chord of Small Stellated Dodecahedron?

The Pentagram Chord of Small Stellated Dodecahedron is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Small Stellated Dodecahedron. It is an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( lc(Pentagram) \) — Pentagram Chord of Small Stellated Dodecahedron (m)
\( TSA \) — Total Surface Area of Small Stellated Dodecahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the pentagram chord based on the total surface area of the small stellated dodecahedron, using mathematical constants and square root functions.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties and proportions of the small stellated dodecahedron, which has applications in mathematical modeling, architecture, and artistic design.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that consists of 12 pentagram faces with five pentagrams meeting at each vertex.

Q2: How is the pentagram chord related to the total surface area?
A: The pentagram chord can be derived from the total surface area through a mathematical relationship that involves the golden ratio and square roots.

Q3: What are typical values for the pentagram chord?
A: The pentagram chord varies depending on the size of the polyhedron, but it is always proportional to the square root of the total surface area.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron due to its unique geometric properties.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most mathematical and engineering applications.