Circumsphere Radius of Truncated Dodecahedron Given Dodecahedral Edge Length Calculator

Formula Used:

\[ r_c = \frac{\sqrt{74 + (30 \times \sqrt{5})}}{4} \times \frac{l_e}{\sqrt{5}} \]

Circumsphere Radius:

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 in polyhedral studies.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{74 + (30 \times \sqrt{5})}}{4} \times \frac{l_e}{\sqrt{5}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( l_e \) — Dodecahedral Edge Length (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the circumsphere radius based on the edge length of the original dodecahedron before truncation.

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 architecture, material science, and mathematical modeling of complex structures.

4. Using the Calculator

Tips: Enter the dodecahedral edge length in meters. The value must be positive and non-zero. The calculator will compute the circumsphere radius using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A Truncated Dodecahedron is an Archimedean solid obtained by cutting the corners of a regular dodecahedron, resulting in 20 equilateral triangles and 12 regular decagons.

Q2: How is this different from a regular dodecahedron's circumsphere?
A: The circumsphere radius of a truncated dodecahedron is larger than that of the original dodecahedron due to the truncation process that moves vertices outward.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, and in understanding the properties of certain molecular structures and crystalline forms.

Q4: How accurate is this formula?
A: The formula is mathematically exact for perfect geometric shapes. Real-world applications may require adjustments for material properties and construction tolerances.

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