Volume of Truncated Dodecahedron given Circumsphere Radius Calculator

Volume of Truncated Dodecahedron Formula:

\[ V = \frac{5}{12} \times (99 + 47\sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{74 + 30\sqrt{5}}} \right)^3 \]

Volume of Truncated Dodecahedron:

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1. What is the Volume of Truncated Dodecahedron?

The Volume of Truncated Dodecahedron is the total quantity of three dimensional space enclosed by the surface of the Truncated Dodecahedron. It is calculated based on the circumsphere radius using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{12} \times (99 + 47\sqrt{5}) \times \left( \frac{4 \times r_c}{\sqrt{74 + 30\sqrt{5}}} \right)^3 \]

Where:

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

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

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in various fields including architecture, engineering, and mathematics. It helps in 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 using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by truncating the vertices of a dodecahedron, resulting in 20 triangular faces, 12 decagonal faces, 60 vertices, and 90 edges.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the truncated dodecahedron such that all vertices lie on the sphere's surface.

Q3: What are typical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, and mathematical research involving polyhedral structures.

Q4: How accurate is this formula?
A: The formula is mathematically exact for perfect truncated dodecahedrons. Real-world accuracy depends on the precision of input measurements.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurements to meters before calculation.