Insphere Radius of Triakis Octahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \frac{6\sqrt{23-16\sqrt{2}}}{(2-\sqrt{2}) \cdot \frac{A}{V}} \cdot \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Insphere Radius (rᵢ):

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1. What is Insphere Radius of Triakis Octahedron?

The Insphere Radius of a Triakis Octahedron is the radius of the sphere that is contained by the Triakis Octahedron in such a way that all the faces are touching the sphere. It represents the largest sphere that can fit inside the polyhedron while being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{6\sqrt{23-16\sqrt{2}}}{(2-\sqrt{2}) \cdot \frac{A}{V}} \cdot \sqrt{\frac{5+2\sqrt{2}}{34}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( \frac{A}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the insphere radius of a Triakis Octahedron based on its surface to volume ratio, incorporating geometric constants specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: The insphere radius is important in geometry and materials science as it helps determine the maximum size of spherical objects that can fit inside a Triakis Octahedron without intersecting its faces. It's also used in packing problems and structural analysis.

4. Using the Calculator

Tips: Enter the surface to volume ratio of the Triakis Octahedron in 1/m. 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 obtained by adding a triangular pyramid to each face of a regular octahedron.

Q2: What units should I use for the surface to volume ratio?
A: The surface to volume ratio should be in reciprocal meters (1/m), where surface area is in square meters and volume is in cubic meters.

Q3: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of values, but extremely small values may approach the limits of floating-point precision.

Q4: What if I get a negative result?
A: The insphere radius should always be positive. If you get a negative result, check that your input value is positive and valid.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula, though floating-point arithmetic may introduce minor rounding errors.