Circumsphere Radius of Stellated Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{6}}{4} \times \frac{\frac{3}{2} \times \sqrt{3}}{\frac{1}{8} \times \sqrt{2} \times \frac{A}{V}} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Stellated Octahedron?

The Circumsphere Radius of a Stellated Octahedron is the radius of the sphere that contains the Stellated Octahedron 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 of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{6}}{4} \times \frac{\frac{3}{2} \times \sqrt{3}}{\frac{1}{8} \times \sqrt{2} \times \frac{A}{V}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( \frac{A}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{6} \) — Square root of 6
\( \sqrt{3} \) — Square root of 3
\( \sqrt{2} \) — Square root of 2

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of the stellated octahedron, incorporating geometric constants and square roots.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial extent of the stellated octahedron, which is important in various fields including crystallography, molecular modeling, and geometric analysis.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Stellated Octahedron?
A: A stellated octahedron is a polyhedron created by extending the faces of a regular octahedron until they intersect, forming a star-shaped solid with triangular pyramids on each face.

Q2: What is Surface to Volume Ratio?
A: Surface to volume ratio is the numerical ratio of the total surface area of a solid to its volume, indicating how much surface area is available per unit volume.

Q3: Why is Circumsphere Radius Important?
A: The circumsphere radius helps determine the smallest sphere that can completely enclose the polyhedron, which is useful in packing problems and spatial analysis.

Q4: Are there Limitations to this Formula?
A: This formula is specifically derived for stellated octahedrons and may not apply to other polyhedral shapes or irregular geometries.

Q5: What Units should be Used?
A: Ensure consistent units - surface to volume ratio in 1/m will yield circumsphere radius in meters. Convert units if necessary for accurate results.