Volume of Triakis Octahedron given Surface to Volume Ratio Calculator

Formula Used:

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

Volume of Triakis Octahedron:

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1. What is the Volume of Triakis Octahedron?

The Volume of Triakis Octahedron is the quantity of three-dimensional space enclosed by the entire surface of the Triakis Octahedron. It is a Catalan solid that can be derived from the octahedron by adding square pyramids to each face.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Triakis Octahedron (m³)
\( \frac{S}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume of a Triakis Octahedron based on its surface to volume ratio, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For the Triakis Octahedron, volume calculation helps in understanding its spatial properties and applications in various fields including crystallography and structural design.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero. The calculator will compute the corresponding volume of the Triakis Octahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Octahedron?
A: A Triakis Octahedron is a Catalan solid that can be obtained by adding square pyramids to each face of a regular octahedron. It has 24 faces, 36 edges, and 14 vertices.

Q2: What are the applications of Triakis Octahedron?
A: Triakis Octahedra appear in crystallography, architecture, and as dice in some tabletop games. Their geometric properties are studied in mathematics and materials science.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The accuracy of the result depends on the precision of the input value.

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

Q5: What units are used in this calculation?
A: The surface to volume ratio is in 1/m and the resulting volume is in cubic meters (m³). Consistent units must be maintained for accurate results.