Circumsphere Radius of Stellated Octahedron given Edge Length of Peaks Calculator

Formula Used:

\[ r_c = \frac{\sqrt{6}}{4} \times (2 \times l_{e(Peaks)}) \]

Circumsphere Radius of Stellated Octahedron:

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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's a fundamental geometric property used in three-dimensional geometry and crystallography.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{6}}{4} \times (2 \times l_{e(Peaks)}) \]

Where:

\( r_c \) — Circumsphere Radius of Stellated Octahedron (m)
\( l_{e(Peaks)} \) — Edge Length of Peaks of Stellated Octahedron (m)
\( \sqrt{6} \) — Square root of 6 (approximately 2.44949)

Explanation: The formula calculates the circumsphere radius based on the edge length of the tetrahedral peaks attached to the faces of the octahedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions of stellated octahedrons, which is important in geometric modeling, crystallography, and materials science applications.

4. Using the Calculator

Tips: Enter the edge length of peaks in meters. 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 attaching tetrahedral pyramids to each face of a regular octahedron, forming a star-like shape.

Q2: How is this different from a regular octahedron's circumsphere?
A: The stellated octahedron has additional vertices from the tetrahedral peaks, resulting in a larger circumsphere radius compared to the base octahedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and geometric art applications.

Q4: Can this formula be used for other stellated polyhedra?
A: No, this specific formula applies only to the stellated octahedron. Other stellated polyhedra have different geometric relationships.

Q5: What units should be used for the input?
A: The calculator uses meters as the default unit, but any consistent unit system can be used as long as the input and output units match.