Pentagram Chord of Great Stellated Dodecahedron Given Volume Calculator

Formula Used:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \left( \frac{4 \times V}{5 \times (3+\sqrt{5})} \right)^{\frac{1}{3}} \]

Pentagram Chord of Great Stellated Dodecahedron:

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

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

2. How Does the Calculator Work?

The calculator uses the formula:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \left( \frac{4 \times V}{5 \times (3+\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( lc(Pentagram) \) — Pentagram Chord of Great Stellated Dodecahedron (m)
\( V \) — Volume of Great Stellated Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the pentagram chord based on the volume of the Great Stellated Dodecahedron, using mathematical constants and cube root operations.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties and spatial relationships within the Great Stellated Dodecahedron, which has applications in mathematical modeling and geometric analysis.

4. Using the Calculator

Tips: Enter the volume of the Great Stellated Dodecahedron in cubic meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, formed by stellating a regular dodecahedron.

Q2: How is the pentagram chord measured?
A: The pentagram chord is measured as the straight-line distance between non-adjacent peak vertices of the pentagram faces.

Q3: What are typical values for the volume?
A: The volume depends on the scale of the polyhedron, but for standard units, it typically ranges from small fractions to larger values depending on the size.

Q4: Are there limitations to this calculation?
A: This calculation assumes a perfect Great Stellated Dodecahedron shape and may not apply to distorted or irregular forms.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Great Stellated Dodecahedron only and may not apply to other geometric shapes.