1. What is the Circumsphere Radius of Great Icosahedron?

The Circumsphere Radius of a Great Icosahedron is the radius of the sphere that contains the polyhedron in such a way that all vertices lie on the sphere's surface. It represents the smallest sphere that can completely enclose the Great Icosahedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

Where:

• \( r_c \) — Circumsphere Radius (m)
• \( TSA \) — Total Surface Area of Great Icosahedron (m²)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
• \( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula derives from the geometric properties of the Great Icosahedron, relating its circumsphere radius to its total surface area through mathematical constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry and 3D modeling for determining the spatial requirements and bounding dimensions of the Great Icosahedron shape. It's particularly important in computational geometry, architectural design, and molecular modeling applications.

4. Using the Calculator

Tips: Enter the total surface area of the Great Icosahedron in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: A Great Icosahedron is a non-convex regular polyhedron and one of the Kepler-Poinsot solids. It has 20 triangular faces that intersect each other.

Q2: How is this different from a regular icosahedron?
A: While both have 20 triangular faces, the Great Icosahedron is star-shaped (non-convex) with faces that intersect, whereas the regular icosahedron is convex with faces that do not intersect.

Q3: What are typical applications of this calculation?
A: This calculation is used in mathematical research, computer graphics, architectural design, and in understanding molecular structures with icosahedral symmetry.

Q4: What units should I use for the input?
A: The calculator expects the total surface area in square meters, but you can use any consistent unit system as long as you interpret the result in the same length units.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the formula. The accuracy depends on the precision of your input value and the computational precision of the system.