Midsphere Radius of Triakis Icosahedron given Insphere Radius Calculator

Formula Used:

\[ r_m = \frac{1+\sqrt{5}}{4} \times \frac{4 \times r_i}{\sqrt{\frac{10 \times (33 + 13 \times \sqrt{5})}{61}}} \]

Midsphere Radius of Triakis Icosahedron:

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1. What is the Midsphere Radius of Triakis Icosahedron?

The Midsphere Radius of Triakis Icosahedron is the radius of the sphere for which all the edges of the Triakis Icosahedron become a tangent line on that sphere. It is an important geometric property of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1+\sqrt{5}}{4} \times \frac{4 \times r_i}{\sqrt{\frac{10 \times (33 + 13 \times \sqrt{5})}{61}}} \]

Where:

\( r_m \) — Midsphere Radius of Triakis Icosahedron (m)
\( r_i \) — Insphere Radius of Triakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the given insphere radius using the geometric properties of the Triakis Icosahedron.

3. Importance of Midsphere Radius Calculation

Details: The midsphere radius is crucial for understanding the geometric properties and spatial relationships of the Triakis Icosahedron. It helps in various applications including 3D modeling, crystallography, and mathematical analysis of polyhedra.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and valid. 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 that is the dual of the truncated dodecahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: How is the midsphere radius different from the insphere radius?
A: The insphere radius is the radius of the sphere inscribed within the polyhedron, touching all faces. The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: What are the applications of this calculation?
A: This calculation is used in geometry, 3D modeling, material science, and architectural design where precise geometric properties are required.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for the Triakis Icosahedron and may not apply to other polyhedra. It assumes perfect geometric proportions.

Q5: What units should be used for input?
A: The calculator uses meters as the default unit, but any consistent unit of length can be used as long as the same unit is used for both input and output.