Midsphere Radius of Rhombic Triacontahedron Given Insphere Radius Calculator

Formula Used:

\[ r_m = \frac{r_i}{\sqrt{\frac{5+2\sqrt{5}}{5}}} \times \frac{5+\sqrt{5}}{5} \]

Midsphere Radius of Rhombic Triacontahedron:

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1. What is the Midsphere Radius of Rhombic Triacontahedron?

The Midsphere Radius of a Rhombic Triacontahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. It represents the sphere that fits perfectly within the polyhedron while touching all its edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{r_i}{\sqrt{\frac{5+2\sqrt{5}}{5}}} \times \frac{5+\sqrt{5}}{5} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( r_i \) — Insphere Radius (m)
\( \sqrt{} \) — Square root function

Explanation: This formula establishes a mathematical relationship between the insphere radius and midsphere radius of a Rhombic Triacontahedron using the golden ratio properties inherent in this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and crystallography for understanding the spatial properties and symmetry of the Rhombic Triacontahedron, which has applications in material science and mathematical modeling.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombic Triacontahedron?
A: A Rhombic Triacontahedron is a convex polyhedron with 30 rhombic faces. It is one of the Catalan solids and is dual to the icosidodecahedron.

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

Q3: What are the applications of this calculation?
A: This calculation is used in geometric modeling, crystallography, and in the study of polyhedral structures in mathematics and materials science.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Rhombic Triacontahedron due to its unique geometric properties.

Q5: What is the significance of the golden ratio in this formula?
A: The Rhombic Triacontahedron has properties related to the golden ratio, which appears in the mathematical relationships between its various measurements.