1. What is the 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 key 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)
• \( TSA \) — Total surface area of icosidodecahedron (m²)

Explanation: The formula calculates the circumsphere radius based on the total surface area of the icosidodecahedron, incorporating the golden ratio and square root functions.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important for understanding the spatial properties of the icosidodecahedron, its relationship with circumscribed spheres, and applications in geometry and 3D modeling.

4. Using the Calculator

Tips: Enter the total surface area in square 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: What are the units for the circumsphere radius?
A: The circumsphere radius is measured in meters (m), consistent with the input surface area units.

Q3: Can this calculator handle different units?
A: The calculator uses the units provided for surface area. Ensure consistent units for accurate results.

Q4: What is the significance of the golden ratio in this formula?
A: The golden ratio \( \frac{1 + \sqrt{5}}{2} \) appears naturally in the geometry of the icosidodecahedron and other polyhedra with pentagonal symmetry.

Q5: Are there other ways to calculate the circumsphere radius?
A: Yes, the circumsphere radius can also be calculated from the edge length or other geometric properties of the icosidodecahedron.