Pentagram Chord Of Great Stellated Dodecahedron Given Circumradius Calculator

Formula Used:

\[ lc(Pentagram) = \frac{(2+\sqrt{5}) \times 4 \times r_c}{\sqrt{3} \times (3+\sqrt{5})} \]

Pentagram Chord of Great Stellated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 represents a key geometric measurement in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ lc(Pentagram) = \frac{(2+\sqrt{5}) \times 4 \times r_c}{\sqrt{3} \times (3+\sqrt{5})} \]

Where:

\( lc(Pentagram) \) — Pentagram Chord of Great Stellated Dodecahedron (m)
\( r_c \) — Circumradius of Great Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula establishes the mathematical relationship between the circumradius of the Great Stellated Dodecahedron and the pentagram chord length, incorporating the golden ratio properties inherent in this geometric structure.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties of the Great Stellated Dodecahedron, which is one of the Kepler-Poinsot polyhedra. This measurement is crucial for architectural applications, mathematical modeling, and studying the symmetry properties of this complex polyhedral form.

4. Using the Calculator

Tips: Enter the circumradius of the Great Stellated Dodecahedron in meters. The value must be positive and non-zero. The calculator will compute the corresponding pentagram chord length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: The Great Stellated Dodecahedron is one of the four Kepler-Poinsot polyhedra, formed by stellating a regular dodecahedron. It features pentagram faces and has a highly symmetric structure.

Q2: How is the circumradius defined for this polyhedron?
A: The circumradius is the radius of the sphere that contains the Great Stellated Dodecahedron such that all vertices lie on the sphere's surface.

Q3: What are typical values for the pentagram chord?
A: The pentagram chord length depends on the size of the polyhedron. For a unit circumradius, the pentagram chord is approximately 1.868 meters.

Q4: Are there practical applications of this calculation?
A: Yes, this calculation is used in architectural design, mathematical education, crystal structure analysis, and the study of geometric patterns in nature and art.

Q5: What mathematical concepts are involved in this formula?
A: The formula incorporates square roots, the golden ratio (φ = (1+√5)/2), and geometric relationships specific to the Great Stellated Dodecahedron's structure.