Volume of Dodecahedron given Circumsphere Radius Calculator

Volume of Dodecahedron Formula:

\[ V = \frac{1}{4} \times (15 + 7\sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^3 \]

Volume of Dodecahedron:

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1. What is the Volume of Dodecahedron?

The volume of a dodecahedron represents the total three-dimensional space enclosed by its twelve pentagonal faces. It is a crucial geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{1}{4} \times (15 + 7\sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Dodecahedron (m³)
\( r_c \) — Circumsphere Radius of Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume of a regular dodecahedron based on the radius of its circumscribed sphere, incorporating mathematical constants and geometric relationships.

3. Importance of Volume Calculation

Details: Calculating the volume of a dodecahedron is essential in geometry, architecture, material science, and various engineering applications where this specific polyhedral shape is utilized.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding volume of the dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a regular polyhedron with twelve identical pentagonal faces, twenty vertices, and thirty edges.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that passes through all vertices of the dodecahedron.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons.

Q4: What are practical applications of dodecahedron volume calculation?
A: Applications include architectural design, crystal structure analysis, game development, and mathematical modeling.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons, with accuracy limited only by the precision of input values and computational rounding.