Surface to Volume Ratio of Great Stellated Dodecahedron given Volume Calculator

Formula Used:

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

Surface to Volume Ratio (SA:V):

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

The Surface to Volume Ratio (SA:V) of a Great 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:

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

Where:

\( V \) — Volume of the Great 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 geometric relationships specific to the Great 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 polyhedra, it helps understand properties like heat transfer, diffusion rates, and structural efficiency.

4. Using the Calculator

Tips: Enter the volume of the Great 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 Great Stellated Dodecahedron?
A: It's a Kepler-Poinsot polyhedron with 12 pentagram faces, 30 edges, and 12 vertices, created by extending the faces of a regular dodecahedron.

Q2: Why is surface to volume ratio important?
A: It indicates how efficiently a shape can exchange materials or energy with its environment. Higher ratios mean more surface area relative to volume.

Q3: What units are used in this calculation?
A: Volume is in cubic meters (m³) and surface to volume ratio is in reciprocal meters (1/m).

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Great Stellated Dodecahedron only. Other polyhedra have different geometric relationships.

Q5: What if I have the surface area instead of volume?
A: You would need a different formula that calculates SA:V directly from surface area, or first calculate volume from surface area if possible.