1. What is Circumsphere Radius of Icosidodecahedron?

The Circumsphere Radius of an Icosidodecahedron is the radius of the sphere that contains the Icosidodecahedron in such a way that all the vertices are lying on the sphere. It is a fundamental geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumsphere Radius of Icosidodecahedron (m)
• \( l_e \) — Edge Length of Icosidodecahedron (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the circumsphere radius based on the edge length using the golden ratio constant \( \frac{1 + \sqrt{5}}{2} \).

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important for understanding the spatial dimensions of the icosidodecahedron, its relationship with circumscribed spheres, and for various applications in geometry, architecture, and material science.

4. Using the Calculator

Tips: Enter the edge length of the icosidodecahedron in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosidodecahedron?
A: An icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 vertices, and 60 edges.

Q2: Why is the golden ratio involved in this calculation?
A: The icosidodecahedron's geometry is closely related to the golden ratio, which appears naturally in its circumsphere radius formula.

Q3: What are the units for the circumsphere radius?
A: The circumsphere radius is expressed in the same units as the edge length (typically meters).

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the icosidodecahedron. Other polyhedra have different circumsphere radius formulas.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using the precise value of the golden ratio. The calculator provides results with 9 decimal places for practical accuracy.