Volume of Great Dodecahedron given Ridge Length Calculator

Volume of Great Dodecahedron Formula:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{2 \times l_{Ridge}}{\sqrt{5} - 1} \right)^3 \]

Volume of Great Dodecahedron:

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1. What is the Volume of Great Dodecahedron?

The Volume of Great Dodecahedron is the total quantity of three dimensional space enclosed by the surface of the Great Dodecahedron. It is a non-convex polyhedron with pentagonal faces and is one of the Kepler-Poinsot solids.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{2 \times l_{Ridge}}{\sqrt{5} - 1} \right)^3 \]

Where:

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

Explanation: This formula calculates the volume of a Great Dodecahedron based on its ridge length, using mathematical constants derived from its geometric properties.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For the Great Dodecahedron, understanding its volume helps in material estimation, structural analysis, and mathematical modeling of complex polyhedra.

4. Using the Calculator

Tips: Enter the ridge length in meters. The value must be positive and non-zero. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: The Great Dodecahedron is one of the four Kepler-Poinsot solids, a non-convex polyhedron with pentagonal faces that intersect each other.

Q2: How is ridge length defined?
A: Ridge Length of Great Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Dodecahedron.

Q3: What are the units of measurement?
A: The calculator uses meters for length and cubic meters for volume. Ensure consistent units for accurate results.

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

Q5: What is the precision of the calculation?
A: The calculator provides results with 6 decimal places precision, suitable for most engineering and mathematical applications.