Total Surface Area Of Hexakis Octahedron Given Short Edge Calculator

Formula Used:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{14 \times le(Short)}{10 - \sqrt{2}} \right)^2 \]

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 Hexakis Octahedron is the amount or quantity of two dimensional space covered on the surface of Hexakis Octahedron. It represents the sum of the areas of all the triangular faces of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{14 \times le(Short)}{10 - \sqrt{2}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( le(Short) \) — Short Edge of Hexakis Octahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the length of the shortest edge of the Hexakis Octahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra like the Hexakis Octahedron is crucial in various fields including materials science, architecture, and 3D modeling. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the length of the short edge in meters. The value must be positive and greater than zero. The calculator will compute the total surface area using the specialized formula for Hexakis Octahedron.

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 cube. It has 48 faces, 72 edges, and 26 vertices.

Q2: Why is the formula so complex?
A: The complexity arises from the geometric properties of the Hexakis Octahedron, which requires precise mathematical relationships between its edges and surface area.

Q3: What units should I use for input?
A: The calculator expects input in meters, and outputs surface area in square meters. You can convert from other units as needed.

Q4: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula, with results rounded to 10 decimal places for practical use.

Q5: Can this calculator handle very large or very small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.