Pentagram Chord of Great Stellated Dodecahedron Given Surface to Volume Ratio Calculator

Formula Used:

\[ lc(Pentagram) = \frac{(2+\sqrt{5}) \times 15 \times \sqrt{5+2\sqrt{5}}}{\frac{5}{4} \times (3+\sqrt{5}) \times SA:V} \]

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 formula:

\[ lc(Pentagram) = \frac{(2+\sqrt{5}) \times 15 \times \sqrt{5+2\sqrt{5}}}{\frac{5}{4} \times (3+\sqrt{5}) \times SA:V} \]

Where:

\( SA:V \) — Surface to Volume Ratio of the Great Stellated Dodecahedron (1/m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the pentagram chord length based on the surface to volume ratio of the polyhedron, incorporating mathematical constants related to the golden ratio and geometric properties of the dodecahedron.

3. Importance of Pentagram Chord Calculation

Details: Calculating the pentagram chord is essential for understanding the geometric properties and proportions of the Great Stellated Dodecahedron, which has applications in mathematical modeling, architectural design, and crystallography studies.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. The value must be positive and greater than zero for accurate calculation.

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 extending the faces of a regular dodecahedron until they intersect.

Q2: How is the surface to volume ratio measured?
A: The surface to volume ratio is calculated as the total surface area divided by the volume of the polyhedron, typically measured in 1/m units.

Q3: What are typical values for SA:V ratio?
A: The SA:V ratio depends on the size and specific dimensions of the polyhedron, with smaller polyhedra generally having higher ratios.

Q4: Are there practical applications of this calculation?
A: Yes, this calculation is used in mathematical research, geometric modeling, and in understanding the properties of complex polyhedral structures.

Q5: What precision should I use for the SA:V input?
A: For accurate results, use at least 4 decimal places when entering the SA:V ratio value.