Total Surface Area of Hexakis Octahedron Calculator

Formula Used:

\[ TSA = (le_{Long})^2 \times \sqrt{543 + 176 \times \sqrt{2}} \times \frac{3}{7} \]

Total Surface Area of Hexakis Octahedron:

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1. What is the Total Surface Area of Hexakis Octahedron?

The Total Surface Area of a Hexakis Octahedron is the total area of all the faces of this complex polyhedron. A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cube.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = (le_{Long})^2 \times \sqrt{543 + 176 \times \sqrt{2}} \times \frac{3}{7} \]

Where:

\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( le_{Long} \) — Long Edge of Hexakis Octahedron (m)

Explanation: This formula calculates the total surface area based on the long edge measurement of the Hexakis Octahedron, incorporating mathematical constants and operations.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is important in geometry, material science, and engineering applications where surface properties and coverage need to be determined for this specific polyhedral shape.

4. Using the Calculator

Tips: Enter the Long Edge of Hexakis Octahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is the formula so complex?
A: The complexity arises from the geometric properties of the Hexakis Octahedron, which requires sophisticated mathematical expressions to accurately calculate its surface area.

Q3: What units should I use for the Long Edge?
A: The calculator uses meters as the default unit, but you can use any consistent unit of length as long as you interpret the surface area result in the corresponding squared units.

Q4: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this formula accurate for all Hexakis Octahedrons?
A: Yes, this formula is mathematically derived and provides accurate results for any regular Hexakis Octahedron when the correct long edge measurement is provided.