Volume Of Great Icosahedron Given Mid Ridge Length Calculator

Volume Of Great Icosahedron Formula:

\[ V = \frac{25 + 9\sqrt{5}}{4} \times \left( \frac{2 \times l_{\text{Ridge(Mid)}}}{1 + \sqrt{5}} \right)^3 \]

Volume Of Great Icosahedron:

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1. What Is The Volume Of Great Icosahedron?

The Great Icosahedron is one of the Kepler-Poinsot polyhedra. It is a non-convex polyhedron with 20 triangular faces. The volume represents the total three-dimensional space enclosed by the surface of the Great Icosahedron.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ V = \frac{25 + 9\sqrt{5}}{4} \times \left( \frac{2 \times l_{\text{Ridge(Mid)}}}{1 + \sqrt{5}} \right)^3 \]

Where:

\( V \) — Volume of Great Icosahedron (m³)
\( l_{\text{Ridge(Mid)}} \) — Mid Ridge Length of Great Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the volume based on the mid ridge length, incorporating the mathematical constant φ (phi) through the expression involving √5.

3. Importance Of Volume Calculation

Details: Calculating the volume of geometric shapes like the Great Icosahedron is essential in various fields including mathematics, architecture, and 3D modeling. It helps in understanding spatial properties and material requirements.

4. Using The Calculator

Tips: Enter the mid ridge length in meters. The value must be positive and greater than zero for accurate calculation.

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 four regular star polyhedra.

Q2: How is the mid ridge length defined?
A: The mid ridge length is the length of any edge that starts from the peak vertex and ends on the interior of the pentagon on which each peak of the Great Icosahedron is attached.

Q3: What units should be used for input?
A: The calculator accepts input in meters, but any consistent unit can be used as long as the output volume is interpreted in cubic units of the input.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Great Icosahedron only. Other polyhedra have different volume formulas.

Q5: What is the significance of √5 in the formula?
A: √5 appears in the formula due to the relationship with the golden ratio φ = (1+√5)/2, which is fundamental to the geometry of icosahedral structures.