Short Edge of Hexakis Octahedron given Total Surface Area Calculator

Formula Used:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \sqrt{\frac{7 \times \text{Total Surface Area}}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Short Edge of Hexakis Octahedron:

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1. What is the Short Edge of Hexakis Octahedron?

The Short Edge of Hexakis Octahedron is the length of the shortest edge of any of the congruent triangular faces of the Hexakis Octahedron. It is an important geometric measurement in this complex polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ \text{Short Edge} = \frac{10 - \sqrt{2}}{14} \times \sqrt{\frac{7 \times \text{Total Surface Area}}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Where:

\( \text{Total Surface Area} \) — The total surface area of the Hexakis Octahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the shortest edge length based on the total surface area of the Hexakis Octahedron, using specific mathematical constants and operations.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is crucial for understanding the geometric properties of Hexakis Octahedron, including its symmetry, volume calculations, and spatial relationships between different edges and faces.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding short edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron. It has 48 faces, 72 edges, and 26 vertices.

Q2: Why is this formula so complex?
A: The formula involves complex mathematical relationships between the surface area and edge lengths due to the intricate geometry of the Hexakis Octahedron.

Q3: What are the units of measurement?
A: The total surface area should be in square meters, and the resulting short edge will be in meters.

Q4: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may affect computational precision.

Q5: Are there other ways to calculate the short edge?
A: Yes, the short edge can also be calculated from other parameters like volume or long edge, but this calculator specifically uses the total surface area.