Circumsphere Radius of Snub Cube given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \sqrt{\frac{3 - C}{4(2 - C)}} \times \frac{2(3 + 4\sqrt{3})}{\frac{RA}{V} \times \frac{3\sqrt{C - 1} + 4\sqrt{C + 1}}{3\sqrt{2 - C}}} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Snub Cube?

The Circumsphere Radius of a Snub Cube is the radius of the sphere that contains the Snub Cube in such a way that all the vertices are lying on the sphere. It is an important geometric property that helps in understanding the spatial dimensions and properties of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ r_c = \sqrt{\frac{3 - C}{4(2 - C)}} \times \frac{2(3 + 4\sqrt{3})}{\frac{RA}{V} \times \frac{3\sqrt{C - 1} + 4\sqrt{C + 1}}{3\sqrt{2 - C}}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( C \) — Tribonacci constant (≈1.839286755214161)
\( \frac{RA}{V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{} \) — Square root function

Explanation: This formula relates the circumsphere radius to the surface-to-volume ratio using the mathematical constant Tribonacci constant, which is specific to the geometry of the snub cube.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial extent of the snub cube, its packing efficiency, and its geometric properties in three-dimensional space. This measurement is particularly important in crystallography, materials science, and geometric modeling.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and valid for accurate calculation of the circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Snub Cube?
A: A snub cube is an Archimedean solid with 38 faces - 6 squares and 32 equilateral triangles. It is a chiral polyhedron, meaning it has two enantiomorphic forms.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161. It appears in various geometric properties of the snub cube.

Q3: How is surface to volume ratio related to circumsphere radius?
A: The surface to volume ratio and circumsphere radius are inversely related through the geometric properties of the snub cube, as expressed in the given formula.

Q4: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the specific dimensions of the snub cube. For a unit snub cube, the circumsphere radius is approximately 1.34371 units.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the snub cube geometry and uses the Tribonacci constant, which is unique to this particular Archimedean solid.