Circumsphere Radius of Snub Cube given Volume Calculator

Formula Used:

\[ r_c = \sqrt{\frac{3 - [Tribonacci_C]}{4 \cdot (2 - [Tribonacci_C])}} \cdot \left( \frac{3 \cdot \sqrt{2 - [Tribonacci_C]} \cdot V}{(3 \cdot \sqrt{[Tribonacci_C] - 1}) + (4 \cdot \sqrt{[Tribonacci_C] + 1})} \right)^{\frac{1}{3}} \]

Circumsphere Radius of Snub Cube (rₑ):

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1. What is the 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 used in three-dimensional geometry and solid modeling.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of snub cube (m³)
\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)

Explanation: This formula relates the circumsphere radius to the volume of the snub cube using the Tribonacci constant, which is characteristic of this particular Archimedean solid.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial properties of the snub cube, its relationship with circumscribed spheres, and for applications in crystallography, molecular modeling, and geometric analysis.

4. Using the Calculator

Tips: Enter the volume of the snub cube in cubic meters. The value must be positive and valid. The calculator will compute the corresponding 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 with 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 accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the snub cube, using the precise value of the Tribonacci constant.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the snub cube due to its unique geometric properties and the involvement of the Tribonacci constant.

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, computer graphics, and architectural design where snub cube geometry is employed.