Circumsphere Radius of Great Stellated Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{3\sqrt{3}\sqrt{5 + 2\sqrt{5}}}{AV} \]

Circumsphere Radius of Great Stellated Dodecahedron:

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1. What is the Circumsphere Radius of Great Stellated Dodecahedron?

The Circumsphere Radius of a Great Stellated Dodecahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices lie on the sphere. It is an important geometric property that helps in understanding the spatial dimensions and symmetry of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{3\sqrt{3}\sqrt{5 + 2\sqrt{5}}}{AV} \]

Where:

\( r_c \) — Circumsphere Radius of Great Stellated Dodecahedron (m)
\( AV \) — SA:V of Great Stellated Dodecahedron (1/m)

Explanation: This formula calculates the circumsphere radius based on the surface area to volume ratio of the Great Stellated Dodecahedron, utilizing mathematical constants and square root functions.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial extent and geometric properties of the Great Stellated Dodecahedron. It helps in various applications including 3D modeling, mathematical analysis, and geometric studies of polyhedra.

4. Using the Calculator

Tips: Enter the SA:V (Surface Area to Volume Ratio) of the Great Stellated Dodecahedron in 1/m. 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: The Great Stellated Dodecahedron is one of the Kepler-Poinsot polyhedra, featuring star-shaped faces and complex symmetry properties.

Q2: How is SA:V ratio related to circumsphere radius?
A: The SA:V ratio provides information about the surface area relative to volume, which inversely relates to the circumsphere radius through the given mathematical formula.

Q3: What are typical values for SA:V ratio?
A: The SA:V ratio varies depending on the size and proportions of the polyhedron, but is always a positive value greater than zero.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to the Great Stellated Dodecahedron. Other polyhedra have different formulas for calculating circumsphere radius.

Q5: What units should I use for the inputs and results?
A: The calculator uses meters for length measurements and 1/m for the SA:V ratio. Ensure consistent units for accurate results.