Total Surface Area Of Triakis Octahedron Given Volume Calculator

Formula Used:

\[ TSA = 6 \times \left( \frac{V}{2-\sqrt{2}} \right)^{2/3} \times \sqrt{23 - (16 \times \sqrt{2})} \]

Total Surface Area of Triakis Octahedron:

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

The Total Surface Area of a Triakis Octahedron is the total quantity of plane enclosed on the entire surface of this polyhedron. It represents the sum of the areas of all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 6 \times \left( \frac{V}{2-\sqrt{2}} \right)^{2/3} \times \sqrt{23 - (16 \times \sqrt{2})} \]

Where:

\( TSA \) — Total Surface Area of Triakis Octahedron (m²)
\( V \) — Volume of Triakis Octahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the total surface area based on the volume of the Triakis Octahedron, using mathematical relationships between volume and surface area for this specific polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, engineering, material science, and 3D modeling. It helps determine material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the volume of the Triakis Octahedron in cubic meters. The value must be positive and greater than zero for accurate calculation.

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 triangular pyramids added to each face, resulting in 24 isosceles triangular faces.

Q2: What are the units for the input and output?
A: The input volume should be in cubic meters (m³), and the output surface area will be in square meters (m²).

Q3: 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.

Q4: Is this formula accurate for all Triakis Octahedrons?
A: Yes, this formula is mathematically derived and provides exact results for any regular Triakis Octahedron when given its volume.

Q5: What if I get an error or unexpected result?
A: Ensure you've entered a valid positive number for the volume. The calculator requires numeric input greater than zero.