1. What is the Midsphere Radius of Deltoidal Hexecontahedron?

The Midsphere Radius of a Deltoidal Hexecontahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. It represents the sphere that touches every edge of the Deltoidal Hexecontahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_m \) — Midsphere Radius of Deltoidal Hexecontahedron (m)
• \( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the midsphere radius and insphere radius of a Deltoidal Hexecontahedron, incorporating the mathematical constant √5.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and architectural design where Deltoidal Hexecontahedron shapes are used. It helps in understanding the spatial properties and proportions of this complex polyhedron.

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 using the mathematical relationship between these two geometric properties.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: A Deltoidal Hexecontahedron is a Catalan solid with 60 deltoid (kite-shaped) faces. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: How is the midsphere radius different from the insphere radius?
A: The insphere radius touches all faces of the polyhedron, while the midsphere radius touches all edges. They represent different spheres associated with the polyhedron.

Q3: What are the practical applications of this calculation?
A: This calculation is used in geometric modeling, crystal structure analysis, architectural design, and mathematical research involving polyhedral geometry.

Q4: Why does the formula contain √5?
A: The square root of 5 appears frequently in formulas related to polyhedra with pentagonal symmetry, which is characteristic of the Deltoidal Hexecontahedron.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Deltoidal Hexecontahedron. Other polyhedra have different mathematical relationships between their midsphere and insphere radii.