Circumsphere Radius Of Rhombicuboctahedron Given Volume Calculator

Formula Used:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \left( \frac{3V}{2(6 + 5\sqrt{2})} \right)^{\frac{1}{3}} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Rhombicuboctahedron?

The Circumsphere Radius of a Rhombicuboctahedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the sphere's surface. It represents the distance from the center of the polyhedron to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \left( \frac{3V}{2(6 + 5\sqrt{2})} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( V \) — Volume of Rhombicuboctahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula derives from the geometric properties of the Rhombicuboctahedron, relating its volume to the radius of its circumscribed sphere through mathematical constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry and 3D modeling for understanding the spatial dimensions of polyhedra, determining bounding spheres for collision detection, and analyzing the geometric properties of complex shapes.

4. Using the Calculator

Tips: Enter the volume of the Rhombicuboctahedron in cubic meters. The volume must be a positive value greater than zero. The calculator will compute the circumsphere radius using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombicuboctahedron?
A: A Rhombicuboctahedron is an Archimedean solid with 8 triangular and 18 square faces, 24 identical vertices, and 48 edges.

Q2: Why is the formula so complex?
A: The formula incorporates mathematical constants and relationships specific to the geometry of the Rhombicuboctahedron, ensuring accurate calculation of the circumsphere radius from the volume.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the volume. For a unit volume Rhombicuboctahedron, the circumsphere radius is approximately 0.95 units.

Q4: Can this calculator handle very large volumes?
A: Yes, the calculator can handle large volume values, though extremely large values may be limited by PHP's floating-point precision.

Q5: Is this calculation applicable to other polyhedra?
A: No, this specific formula is derived exclusively for the Rhombicuboctahedron. Other polyhedra have different formulas for calculating circumsphere radius from volume.