Volume Of Great Icosahedron Given Circumsphere Radius Calculator

Formula Used:

\[ V = \frac{25 + 9\sqrt{5}}{4} \times \left( \frac{4r_c}{\sqrt{50 + 22\sqrt{5}}} \right)^3 \]

Volume of Great Icosahedron:

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1. What is Volume of Great Icosahedron?

The Volume of Great Icosahedron refers to the total three-dimensional space enclosed by the surface of this complex polyhedron. It is a mathematical measure of the capacity of the Great Icosahedron shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{25 + 9\sqrt{5}}{4} \times \left( \frac{4r_c}{\sqrt{50 + 22\sqrt{5}}} \right)^3 \]

Where:

\( V \) — Volume of Great Icosahedron (m³)
\( r_c \) — Circumsphere Radius of Great Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume based on the circumsphere radius, incorporating mathematical constants and geometric relationships specific to the Great Icosahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes like the Great Icosahedron is fundamental in mathematics, engineering, and architectural design. It helps in understanding spatial properties and material requirements for construction.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is a non-convex uniform polyhedron composed of 20 equilateral triangular faces. It is one of the Kepler-Poinsot solids.

Q2: What is Circumsphere Radius?
A: Circumsphere Radius is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise when using the exact formula with proper input values.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters first or adjust the result accordingly.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical research, geometric modeling, architectural design, and educational purposes to understand complex polyhedral structures.