Edge Length of Stellated Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ Edge\ Length = \frac{\frac{3}{2} \times \sqrt{3}}{\frac{1}{8} \times \sqrt{2} \times Surface\ to\ Volume\ Ratio} \]

Edge Length of Stellated Octahedron:

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1. What is the Edge Length of Stellated Octahedron?

The Edge Length of a Stellated Octahedron is the distance between any pair of adjacent peak vertices of this complex polyhedron. It is a fundamental geometric property that determines the overall size and proportions of the shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ Edge\ Length = \frac{\frac{3}{2} \times \sqrt{3}}{\frac{1}{8} \times \sqrt{2} \times Surface\ to\ Volume\ Ratio} \]

Where:

\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{2} \) — Square root of 2 (approximately 1.414)
Surface to Volume Ratio — The ratio of surface area to volume of the stellated octahedron

Explanation: This formula derives from the geometric relationships between the edge length, surface area, and volume of a stellated octahedron.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the dimensional properties of stellated octahedrons, which are important in crystallography, molecular modeling, and architectural design where this geometric form is utilized.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m (reciprocal meters). The value must be greater than zero. The calculator will compute the corresponding edge length in meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Stellated Octahedron?
A: A stellated octahedron is a complex polyhedron formed by extending the faces of a regular octahedron until they intersect, creating a star-shaped polyhedron with 14 faces.

Q2: How is the surface to volume ratio measured?
A: The surface to volume ratio is calculated by dividing the total surface area of the stellated octahedron by its volume, typically expressed in units of 1/m (reciprocal meters).

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the stellated octahedron. Smaller objects have higher ratios, while larger objects have lower ratios.

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

Q5: What are practical applications of this calculation?
A: This calculation is useful in materials science, nanotechnology, and geometric modeling where understanding the relationship between surface properties and dimensional characteristics is important.