Circumsphere Radius of Rhombicuboctahedron given Total Surface Area Calculator

Formula Used:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \sqrt{\frac{TSA}{2(9 + \sqrt{3})}} \]

Circumsphere Radius (r_c):

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1. What is the 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 is a key geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{5 + 2\sqrt{2}}}{2} \times \sqrt{\frac{TSA}{2(9 + \sqrt{3})}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( TSA \) — Total Surface Area of Rhombicuboctahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the circumsphere radius from the total surface area using the geometric properties specific to the rhombicuboctahedron shape.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the polyhedron, which is crucial in fields like crystallography, architecture, and 3D modeling where precise geometric measurements are required.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the circumsphere radius based on the provided surface area.

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 circumsphere radius important?
A: It helps determine the smallest sphere that can completely enclose the polyhedron, which is useful in packaging, containment, and spatial analysis problems.

Q3: What units should I use for input?
A: The calculator expects the total surface area in square meters, and returns the circumsphere radius in meters.

Q4: Are there limitations to this formula?
A: This formula is specifically derived for the regular rhombicuboctahedron and may not apply to modified or irregular versions of the shape.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the rhombicuboctahedron. Other polyhedra have different formulas for calculating their circumsphere radii.