Pentagram Chord of Small Stellated Dodecahedron given Pyramidal Height Calculator

Formula Used:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \frac{5 \times h_{Pyramid}}{\sqrt{25+10 \times \sqrt{5}}} \]

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 represents a key geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ lc(Pentagram) = (2+\sqrt{5}) \times \frac{5 \times h_{Pyramid}}{\sqrt{25+10 \times \sqrt{5}}} \]

Where:

\( lc(Pentagram) \) — Pentagram Chord of Small Stellated Dodecahedron (meters)
\( h_{Pyramid} \) — Pyramidal Height of Small Stellated Dodecahedron (meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the pentagram chord based on the pyramidal height, using the mathematical relationship derived from the geometry of the Small Stellated Dodecahedron.

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 pyramidal height in 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 formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron.

Q2: What is the significance of the pentagram chord?
A: The pentagram chord helps define the overall structure and symmetry of the polyhedron, and is important for various geometric calculations and constructions.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula, though practical measurements may have some degree of error.

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

Q5: What units should be used for input?
A: The calculator expects input in meters, though the formula is unit-consistent as long as the same unit is used throughout.