Circumsphere Radius of Dodecahedron Given Face Perimeter Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \cdot (1 + \sqrt{5})}{20} \cdot P_{Face} \]

Circumsphere Radius of Dodecahedron:

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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 is a fundamental geometric property of this regular polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{3} \cdot (1 + \sqrt{5})}{20} \cdot P_{Face} \]

Where:

\( r_c \) — Circumsphere Radius of Dodecahedron (m)
\( P_{Face} \) — Face Perimeter of Dodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula relates the circumsphere radius to the face perimeter through a constant factor derived from the geometric properties of the dodecahedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry, 3D modeling, and various engineering applications where the spatial dimensions of a dodecahedron need to be determined.

4. Using the Calculator

Tips: Enter the face perimeter of the dodecahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Dodecahedron?
A: A dodecahedron is a regular polyhedron with twelve regular pentagonal faces, thirty edges, and twenty vertices.

Q2: How is Face Perimeter related to Circumsphere Radius?
A: The circumsphere radius is directly proportional to the face perimeter through a constant factor derived from the golden ratio and square roots.

Q3: What are the practical applications of this calculation?
A: This calculation is used in geometry, architecture, molecular modeling, and various fields where dodecahedral structures are encountered.

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

Q5: What is the accuracy of this calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons, with precision limited only by input measurement accuracy.