Surface to Volume Ratio of Small Stellated Dodecahedron given Ridge Length Calculator

Formula Used:

\[ SA:V = \frac{15 \times \sqrt{5 + 2\sqrt{5}}}{\frac{5}{4} \times (7 + 3\sqrt{5})} \times \frac{1 + \sqrt{5}}{2 \times l_{Ridge}} \]

Surface to Volume Ratio (SA:V):

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1. What is 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 relationship between the total surface area and the volume of this polyhedron. It's an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SA:V = \frac{15 \times \sqrt{5 + 2\sqrt{5}}}{\frac{5}{4} \times (7 + 3\sqrt{5})} \times \frac{1 + \sqrt{5}}{2 \times l_{Ridge}} \]

Where:

\( l_{Ridge} \) — Ridge Length of Small Stellated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the surface to volume ratio based on the ridge length of the polyhedron, incorporating mathematical constants specific to the geometry of the Small Stellated Dodecahedron.

3. Importance of Surface to Volume Ratio

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and physics. It affects properties like heat transfer, reaction rates, and structural efficiency.

4. Using the Calculator

Tips: Enter the ridge 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: It's a Kepler-Poinsot polyhedron that consists of 12 pentagram faces with 5 pentagrams meeting at each vertex.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which is critical for processes involving surface interactions.

Q3: What units are used in this calculation?
A: Ridge length is in meters (m) and the resulting SA:V ratio is in meters⁻¹ (m⁻¹).

Q4: Can this calculator handle very small ridge lengths?
A: Yes, but extremely small values may require higher precision calculations.

Q5: Is this formula specific to the Small Stellated Dodecahedron?
A: Yes, this formula is specifically derived for the geometric properties of the Small Stellated Dodecahedron.