Surface to Volume Ratio of Small Stellated Dodecahedron given Pentagram Chord Calculator

Formula Used:

\[ SA:V = \frac{15 \cdot \sqrt{5 + 2 \cdot \sqrt{5}}}{\frac{5}{4} \cdot (7 + 3 \cdot \sqrt{5})} \cdot \frac{2 + \sqrt{5}}{l_c(Pentagram)} \]

Surface to Volume Ratio (SA:V):

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1. What is the Surface to Volume Ratio of Small Stellated Dodecahedron?

The Surface to Volume Ratio (SA:V) of a Small Stellated Dodecahedron is a geometric property that represents the ratio of its total surface area to its volume. It's an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SA:V = \frac{15 \cdot \sqrt{5 + 2 \cdot \sqrt{5}}}{\frac{5}{4} \cdot (7 + 3 \cdot \sqrt{5})} \cdot \frac{2 + \sqrt{5}}{l_c(Pentagram)} \]

Where:

\( l_c(Pentagram) \) — Pentagram Chord of Small Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the surface to volume ratio based on the pentagram chord length, incorporating mathematical constants and geometric relationships specific to the small stellated dodecahedron.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. For geometric shapes like the small stellated dodecahedron, it helps understand the relationship between surface area and volume, which is important in optimization problems and structural analysis.

4. Using the Calculator

Tips: Enter the pentagram chord length 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: A Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that represents the first stellation of the dodecahedron, featuring pentagrammic faces.

Q2: What is the Pentagram Chord in this context?
A: The pentagram chord is the distance between any pair of non-adjacent peak vertices of the pentagram corresponding to the Small Stellated Dodecahedron.

Q3: What are typical values for the surface to volume ratio?
A: The ratio depends on the pentagram chord length. Smaller chord lengths result in higher ratios, while larger chord lengths yield lower ratios.

Q4: What units are used in this calculation?
A: The pentagram chord is measured in meters (m), and the surface to volume ratio is expressed in reciprocal meters (m⁻¹).

Q5: Are there any limitations to this formula?
A: This formula is specifically designed for the geometric properties of a perfect small stellated dodecahedron and assumes ideal mathematical conditions.