Circumsphere Radius of Great Icosahedron given Short Ridge Length Calculator

Formula Used:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times \frac{5 \times l_{\text{Ridge(Short)}}}{\sqrt{10}} \]

Circumsphere Radius of Great Icosahedron:

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1. What is Circumsphere Radius of Great Icosahedron?

The Circumsphere Radius of Great Icosahedron is the radius of the sphere that contains the Great Icosahedron in such a way that all the peak vertices are lying on the sphere. It's a fundamental geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{50 + 22\sqrt{5}}}{4} \times \frac{5 \times l_{\text{Ridge(Short)}}}{\sqrt{10}} \]

Where:

\( r_c \) — Circumsphere Radius of Great Icosahedron (m)
\( l_{\text{Ridge(Short)}} \) — Short Ridge Length of Great Icosahedron (m)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula calculates the circumsphere radius based on the short ridge length measurement of the Great Icosahedron, incorporating the golden ratio properties inherent in this geometric shape.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and geometric properties of the Great Icosahedron. It's particularly important in fields of geometry, architecture, and 3D modeling where precise measurements of complex polyhedra are required.

4. Using the Calculator

Tips: Enter the Short Ridge Length measurement in meters. The value must be positive and greater than zero. The calculator will automatically compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra, featuring 20 triangular faces that intersect each other, creating a complex star polyhedron.

Q2: How is Short Ridge Length defined?
A: Short Ridge Length is defined as the maximum vertical distance between the finished bottom level and the finished top height directly above of Great Icosahedron.

Q3: What units should I use for input?
A: The calculator accepts meters as input units, but you can use any consistent unit system as the relationship is proportional.

Q4: Are there practical applications of this calculation?
A: Yes, this calculation is used in architectural design, molecular modeling, and mathematical research involving complex polyhedral structures.

Q5: How accurate is this formula?
A: The formula is mathematically exact for the ideal Great Icosahedron geometry, providing precise results for theoretical calculations.