Circumsphere Radius Of Dodecahedron Given Volume Calculator

Formula Used:

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

Circumsphere Radius (r_c):

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1. What is the Circumsphere Radius of Dodecahedron?

The circumsphere radius of a dodecahedron is the radius of the sphere that contains the dodecahedron in such a way that all the vertices are lying on the sphere. It represents the distance from the center of the dodecahedron to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_c \) — Circumsphere radius (m)
\( V \) — Volume of dodecahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: The formula derives the circumsphere radius from the volume of a regular dodecahedron using mathematical relationships between geometric properties.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, 3D modeling, and various engineering applications where precise spatial relationships of dodecahedral structures need to be determined.

4. Using the Calculator

Tips: Enter the volume of the dodecahedron in cubic meters. The volume must be a positive value greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular dodecahedron?
A: A regular dodecahedron is a three-dimensional shape with 12 identical regular pentagonal faces, 20 vertices, and 30 edges.

Q2: How is the circumsphere radius related to other dodecahedron properties?
A: The circumsphere radius is mathematically related to the volume, edge length, and other geometric properties of a regular dodecahedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and various fields involving geometric analysis.

Q4: Are there limitations to this formula?
A: This formula applies only to regular dodecahedrons. For irregular dodecahedrons, different calculation methods are required.

Q5: Can this calculator handle different units?
A: The calculator uses cubic meters for volume input. For other units, convert your volume measurement to cubic meters before calculation.