Circumsphere Radius of Truncated Dodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{\sqrt{74+(30\times\sqrt{5})} \times 3 \times (\sqrt{3}+(6\times\sqrt{5+(2\times\sqrt{5})}))}{RA/V \times (99+(47\times\sqrt{5}))} \]

Circumsphere Radius (r_c):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Circumsphere Radius of Truncated Dodecahedron?

The Circumsphere Radius of a Truncated Dodecahedron is the radius of the sphere that contains the Truncated Dodecahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{74+(30\times\sqrt{5})} \times 3 \times (\sqrt{3}+(6\times\sqrt{5+(2\times\sqrt{5})}))}{RA/V \times (99+(47\times\sqrt{5}))} \]

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 based on the surface to volume ratio of a truncated dodecahedron, incorporating various mathematical constants and operations.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial properties of truncated dodecahedrons, which have applications in geometry, crystallography, and architectural design.

4. Using the Calculator

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

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 regular triangular faces and 12 regular decagonal faces.

Q2: What are typical values for Surface to Volume Ratio?
A: The surface to volume ratio varies depending on the size and proportions of the truncated dodecahedron, but typically ranges from 0.1 to 10 1/m for most practical applications.

Q3: What units are used in this calculation?
A: The surface to volume ratio is measured in 1/m (inverse meters) and the resulting circumsphere radius is in meters (m).

Q4: Are there limitations to this formula?
A: This formula is specifically designed for perfect truncated dodecahedrons and assumes ideal geometric proportions.

Q5: What practical applications does this calculation have?
A: This calculation is used in various fields including materials science, architectural design, and mathematical modeling of complex geometric structures.