Circumsphere Radius of Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \cdot (1+\sqrt{5}) \cdot 3\sqrt{25+(10\sqrt{5})}}{(RA/V) \cdot (15+(7\sqrt{5}))} \]

Circumsphere Radius (r_c):

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1. What is 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:

\[ r_c = \frac{\sqrt{3} \cdot (1+\sqrt{5}) \cdot 3\sqrt{25+(10\sqrt{5})}}{(RA/V) \cdot (15+(7\sqrt{5}))} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the circumsphere radius of a regular dodecahedron based on its surface to volume ratio, incorporating mathematical constants and geometric relationships specific to dodecahedrons.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, crystallography, and material science for understanding the spatial dimensions and packing efficiency of dodecahedral structures.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and non-zero for accurate calculation.

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 surface to volume ratio defined for a dodecahedron?
A: The surface to volume ratio is calculated by dividing the total surface area by the volume of the dodecahedron.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the size of the dodecahedron. For a unit dodecahedron, the circumsphere radius is approximately 1.401258538.

Q4: Can this formula be used for irregular dodecahedrons?
A: No, this formula is specifically derived for regular dodecahedrons where all faces are identical regular pentagons.

Q5: What practical applications does this calculation have?
A: This calculation is used in various fields including mathematics education, 3D modeling, architectural design, and the study of crystalline structures.