Chord Length of Pentagram given Area and Short Chord Slice Calculator

Formula Used:

\[ l_c = \sqrt{\frac{2A}{\sqrt{5(5-2\sqrt{5})}}} + (l_{Short} \times \phi) \]

Chord Length of Pentagram:

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1. What is the Chord Length of Pentagram?

The Chord Length of Pentagram is the diagonal length of a regular pentagon from which the Pentagram is constructed using its diagonals. It represents the longest straight line that can be drawn between two non-adjacent vertices of the pentagram.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_c = \sqrt{\frac{2A}{\sqrt{5(5-2\sqrt{5})}}} + (l_{Short} \times \phi) \]

Where:

\( l_c \) — Chord Length of Pentagram (m)
\( A \) — Area of Pentagram (m²)
\( l_{Short} \) — Short Chord Slice of Pentagram (m)
\( \phi \) — Golden ratio (≈1.618)

Explanation: The formula combines the geometric relationship between the area and the chord length with the golden ratio property inherent in pentagram geometry.

3. Importance of Chord Length Calculation

Details: Calculating the chord length is essential for geometric analysis, architectural design, and artistic applications involving pentagrams. It helps in understanding the proportional relationships within this sacred geometric shape.

4. Using the Calculator

Tips: Enter the area of the pentagram in square meters and the short chord slice length in meters. Both values must be positive numbers for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the golden ratio (φ) in pentagram geometry?
A: The golden ratio (approximately 1.618) appears naturally in pentagram geometry, where the ratio of the chord length to the side length equals φ.

Q2: How is the short chord slice related to the pentagram?
A: The short chord slice is the edge length of the regular pentagon that forms inside the pentagram when all chords are drawn, creating the characteristic star shape.

Q3: Can this formula be used for any pentagram?
A: This formula applies specifically to regular pentagrams constructed from regular pentagons, where all sides and angles are equal.

Q4: What are practical applications of pentagram chord length calculations?
A: Applications include architectural design, artistic compositions, sacred geometry studies, and mathematical research in geometric properties.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for ideal regular pentagrams, though real-world measurements may introduce slight variations due to measurement precision.