Insphere Radius of Triakis Icosahedron given Midsphere Radius Calculator

Formula Used:

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

Insphere Radius of Triakis Icosahedron:

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1. What is Insphere Radius of Triakis Icosahedron?

The Insphere Radius of Triakis Icosahedron is the radius of the sphere that is contained by the Triakis Icosahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

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

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important for understanding the internal geometry of the Triakis Icosahedron and its relationship with other geometric properties. It helps in various applications including material science, crystallography, and architectural design.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere 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 isosceles triangular faces.

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

Q3: What are typical values for these radii?
A: The values depend on the specific dimensions of the Triakis Icosahedron. Both radii are proportional to the edge length of the polyhedron.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Triakis Icosahedron. Other polyhedra have different formulas for their insphere radii.

Q5: What precision does the calculator provide?
A: The calculator provides results with up to 12 decimal places for accuracy in mathematical and engineering applications.