Circumsphere Radius of Truncated Rhombohedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{14-2\sqrt{5}}}{4} \times \frac{3\sqrt{5+2\sqrt{5}} + 5\sqrt{3} - 2\sqrt{15}}{2 \times \frac{A}{V} \times \frac{5}{3} \times \sqrt{\sqrt{5}-2}} \]

Circumsphere Radius (rₑ):

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1. What is Circumsphere Radius of Truncated Rhombohedron?

The Circumsphere Radius of a Truncated Rhombohedron is the radius of the sphere that contains the polyhedron in such a way that all the vertices are lying on the sphere. 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 formula:

\[ r_c = \frac{\sqrt{14-2\sqrt{5}}}{4} \times \frac{3\sqrt{5+2\sqrt{5}} + 5\sqrt{3} - 2\sqrt{15}}{2 \times \frac{A}{V} \times \frac{5}{3} \times \sqrt{\sqrt{5}-2}} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( \frac{A}{V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (≈2.236)
\( \sqrt{3} \) — Square root of 3 (≈1.732)
\( \sqrt{15} \) — Square root of 15 (≈3.873)

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of a truncated rhombohedron, incorporating various mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in crystallography, materials science, and geometric modeling where truncated rhombohedra appear. It helps in understanding the spatial dimensions and packing efficiency of these structures.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and valid for a truncated rhombohedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated rhombohedron?
A: A truncated rhombohedron is a polyhedron obtained by cutting the corners of a rhombohedron, resulting in a shape with both hexagonal and triangular faces.

Q2: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the specific dimensions of the truncated rhombohedron, but typically ranges from 0.1 to 10 m⁻¹ for most practical applications.

Q3: Where are truncated rhombohedra commonly found?
A: They appear in crystallography (certain crystal structures), architecture, and in some natural mineral formations.

Q4: What are the units of measurement?
A: Surface to volume ratio is measured in m⁻¹, and the resulting circumsphere radius is in meters (m).

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for an ideal truncated rhombohedron, but real-world applications may have slight variations due to manufacturing tolerances or measurement errors.