Surface To Volume Ratio Of Great Stellated Dodecahedron Given Total Surface Area Calculator

Formula Used:

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

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 is available per unit volume of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area of the Great Stellated Dodecahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the surface to volume ratio based on the known mathematical properties of the Great Stellated Dodecahedron and its total surface area.

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 diffusion rates, heat transfer, and structural efficiency.

4. Using the Calculator

Tips: Enter the total surface area of the Great Stellated Dodecahedron in square meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Stellated Dodecahedron?
A: It's one of the Kepler-Poinsot polyhedra, a non-convex regular polyhedron with pentagrammic faces.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available relative to the volume, which affects properties like reactivity, heat dissipation, and structural strength.

Q3: What units are used for this calculation?
A: The surface area should be in square meters (m²), and the resulting ratio will be in reciprocal meters (1/m).

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron due to its unique geometric properties.

Q5: What if I have the volume instead of surface area?
A: You would need a different formula that calculates SA:V directly from volume. This calculator specifically uses total surface area as input.