Pentagram Chord Of Small Stellated Dodecahedron Given Volume Calculator

Formula Used:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \left(\frac{4 \times V}{5 \times (7+3\sqrt{5})}\right)^{\frac{1}{3}} \]

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 represents a key geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \left(\frac{4 \times V}{5 \times (7+3\sqrt{5})}\right)^{\frac{1}{3}} \]

Where:

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

Explanation: The formula derives the pentagram chord length from the volume of the Small Stellated Dodecahedron using geometric relationships and mathematical constants.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties of the Small Stellated Dodecahedron, including its symmetry, proportions, and spatial relationships between vertices.

4. Using the Calculator

Tips: Enter the volume of the Small Stellated Dodecahedron in cubic 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 represents one of the four regular star polyhedra, formed by extending the faces of a regular dodecahedron.

Q2: How is the pentagram chord related to the volume?
A: The pentagram chord can be derived from the volume through a mathematical relationship that accounts for the geometric properties and symmetry of the polyhedron.

Q3: What units should be used for volume input?
A: The calculator expects volume input in cubic meters (m³), and returns the pentagram chord in meters (m).

Q4: Are there any limitations to this calculation?
A: The formula assumes a perfect geometric shape and may not account for manufacturing tolerances or imperfections in physical models.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron and its pentagram chord measurement.