Pyramidal Edge Length Of Triakis Octahedron Given Insphere Radius Calculator

Formula Used:

\[ l_{pyramid} = (2-\sqrt{2}) \times \frac{r_i}{\sqrt{\frac{5+2\sqrt{2}}{34}}} \]

Pyramidal Edge Length of Triakis Octahedron:

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1. What is Pyramidal Edge Length of Triakis Octahedron?

The Pyramidal Edge Length of Triakis Octahedron is the length of the line connecting any two adjacent vertices of pyramid of Triakis Octahedron. It's an important geometric measurement in understanding the structure and properties of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{pyramid} = (2-\sqrt{2}) \times \frac{r_i}{\sqrt{\frac{5+2\sqrt{2}}{34}}} \]

Where:

\( l_{pyramid} \) — Pyramidal Edge Length of Triakis Octahedron (m)
\( r_i \) — Insphere Radius of Triakis Octahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula establishes the mathematical relationship between the insphere radius and the pyramidal edge length of a Triakis Octahedron, incorporating the geometric properties of this specific polyhedron.

3. Importance of Pyramidal Edge Length Calculation

Details: Calculating the pyramidal edge length is crucial for understanding the geometric properties, surface area, volume, and other dimensional characteristics of the Triakis Octahedron. It's particularly important in crystallography, material science, and geometric modeling applications.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding pyramidal edge length using the established geometric relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Octahedron?
A: A Triakis Octahedron is a Catalan solid that can be seen as an octahedron with a triangular pyramid added to each face. It has 24 isosceles triangular faces.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can be contained within the Triakis Octahedron such that it touches all faces.

Q3: Are there any limitations to this formula?
A: This formula is specifically derived for the Triakis Octahedron and assumes perfect geometric proportions. It may not apply to distorted or irregular variations.

Q4: What units should be used?
A: The formula works with consistent units. Typically meters are used, but any length unit can be used as long as it's consistent for both input and output.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the Triakis Octahedron. The accuracy depends on the precision of the input value.