Surface to Volume Ratio of Small Stellated Dodecahedron given Volume Calculator

Formula Used:

\[ SA:V = \frac{15 \cdot \sqrt{5 + 2\sqrt{5}}}{\frac{5}{4} \cdot (7 + 3\sqrt{5})} \cdot \left( \frac{5 \cdot (7 + 3\sqrt{5})}{4 \cdot V} \right)^{\frac{1}{3}} \]

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 the numerical ratio of its total surface area to its volume. It's an important geometric property that describes how much surface area the polyhedron has relative to its volume.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SA:V = \frac{15 \cdot \sqrt{5 + 2\sqrt{5}}}{\frac{5}{4} \cdot (7 + 3\sqrt{5})} \cdot \left( \frac{5 \cdot (7 + 3\sqrt{5})}{4 \cdot V} \right)^{\frac{1}{3}} \]

Where:

\( V \) — Volume of the Small Stellated Dodecahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the surface to volume ratio based on the given volume of the polyhedron, using mathematical constants and operations specific to the geometry of 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 engineering. It helps understand properties like heat transfer, chemical reactivity, and structural efficiency of geometric shapes.

4. Using the Calculator

Tips: Enter the volume of the Small Stellated Dodecahedron in cubic 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 intersecting each other, creating a star-shaped three-dimensional figure.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which affects properties like diffusion rates, heat dissipation, and chemical reactivity.

Q3: What units are used in this calculation?
A: The volume is in cubic meters (m³) and the surface to volume ratio is in per meter (m⁻¹).

Q4: Can this calculator handle very small volumes?
A: Yes, but extremely small volumes may result in very large surface to volume ratios due to the inverse relationship.

Q5: Is this formula specific to Small Stellated Dodecahedron?
A: Yes, this formula is derived specifically for the geometric properties of the Small Stellated Dodecahedron and may not apply to other polyhedra.