Midsphere Radius of Triakis Octahedron Given Volume Calculator

Formula Used:

\[ r_m = \frac{1}{2} \times \left( \frac{V}{2 - \sqrt{2}} \right)^{\frac{1}{3}} \]

Midsphere Radius of Triakis Octahedron (rₘ):

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1. What is Midsphere Radius of Triakis Octahedron?

The Midsphere Radius of Triakis Octahedron is the radius of the sphere for which all the edges of the Triakis Octahedron become a tangent line on that sphere. It represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1}{2} \times \left( \frac{V}{2 - \sqrt{2}} \right)^{\frac{1}{3}} \]

Where:

\( r_m \) — Midsphere radius of Triakis Octahedron (m)
\( V \) — Volume of Triakis Octahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the midsphere radius from the volume of a Triakis Octahedron, using the mathematical relationship between volume and midsphere radius for this specific polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of Triakis Octahedrons. It helps in determining the sphere that would perfectly fit tangent to all edges of the polyhedron.

4. Using the Calculator

Tips: Enter the volume of the Triakis Octahedron in cubic meters. The volume must be a positive value 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.

Q2: How is this different from insphere radius?
A: The midsphere radius touches the midpoints of edges, while the insphere radius is tangent to the faces of the polyhedron.

Q3: What units should I use for volume?
A: The calculator uses cubic meters (m³), but you can use any consistent unit as long as the result is interpreted in the same unit.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to Triakis Octahedrons. Other polyhedrons have different relationships between volume and midsphere radius.

Q5: What if I have the edge length instead of volume?
A: You would need to first calculate the volume from the edge length using the appropriate formula for Triakis Octahedrons before using this calculator.