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Midsphere Radius of Triakis Icosahedron given Volume Calculator

Formula Used:

\[ r_m = \frac{1+\sqrt{5}}{4} \times \left( \frac{44 \times V}{5 \times (5 + 7\sqrt{5})} \right)^{1/3} \]

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1. What is the Midsphere Radius of Triakis Icosahedron?

The midsphere radius of a Triakis Icosahedron is the radius of the sphere that is tangent to all edges of the polyhedron. For a Triakis Icosahedron, which is an icosahedron with triangular pyramids added to each face, this sphere touches each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1+\sqrt{5}}{4} \times \left( \frac{44 \times V}{5 \times (5 + 7\sqrt{5})} \right)^{1/3} \]

Where:

Explanation: This formula derives from the geometric properties of the Triakis Icosahedron, relating its midsphere radius to its volume through the golden ratio and cubic root relationships.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the Triakis Icosahedron, including its symmetry and packing characteristics.

4. Using the Calculator

Tips: Enter the volume of the Triakis Icosahedron in cubic meters. The volume must be a positive value. The calculator will compute the corresponding midsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid created by adding a triangular pyramid to each face of a regular icosahedron, resulting in a polyhedron with 60 isosceles triangular faces.

Q2: How is the midsphere different from the insphere?
A: The midsphere is tangent to all edges, while the insphere is tangent to all faces. For most polyhedra, these are different spheres with different radii.

Q3: What are typical values for the midsphere radius?
A: The midsphere radius depends on the volume. For a given volume, the midsphere radius of a Triakis Icosahedron is determined by its specific geometric proportions.

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

Q5: What precision does the calculator provide?
A: The calculator provides results with up to 12 decimal places for accuracy in geometric calculations.

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