Volume of Great Dodecahedron Given Circumsphere Radius Calculator

Formula Used:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{4 \times r_c}{\sqrt{10 + (2 \times \sqrt{5})}} \right)^3 \]

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.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{4} \times (\sqrt{5} - 1) \times \left( \frac{4 \times r_c}{\sqrt{10 + (2 \times \sqrt{5})}} \right)^3 \]

Where:

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

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

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes like the Great Dodecahedron is crucial in fields such as mathematics, architecture, and 3D modeling for understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the volume based on the provided radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: The Great Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagonal faces. It is non-convex and has intersecting faces.

Q2: How is the circumsphere radius defined?
A: The circumsphere radius is the radius of the sphere that contains the Great Dodecahedron such that all vertices lie on the sphere.

Q3: What are typical values for the circumsphere radius?
A: The circumsphere radius depends on the size of the Great Dodecahedron. It can vary widely based on the specific dimensions.

Q4: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Great Dodecahedron. Other polyhedra have their own unique volume formulas.

Q5: What units should be used?
A: The calculator uses meters for input and cubic meters for output. Ensure consistent units for accurate results.