Circumsphere Radius of Great Dodecahedron given Pyramidal Height Calculator

Formula Used:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3 - \sqrt{5})} \]

Circumsphere Radius of Great Dodecahedron:

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1. What is Circumsphere Radius of Great Dodecahedron?

The Circumsphere Radius of a Great Dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It is a key geometric property in three-dimensional geometry.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \frac{6 \times h_{Pyramid}}{\sqrt{3} \times (3 - \sqrt{5})} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( h_{Pyramid} \) — Pyramidal Height of Great Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (≈2.23607)
\( \sqrt{3} \) — Square root of 3 (≈1.73205)

Explanation: This formula calculates the circumsphere radius based on the pyramidal height of the Great Dodecahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and geometric properties of the Great Dodecahedron, particularly in fields like crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the Pyramidal Height of the Great Dodecahedron in meters. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagonal faces that intersect each other.

Q2: How is Pyramidal Height defined?
A: Pyramidal Height refers to the height of the inward-directed tetrahedral pyramids that form the Great Dodecahedron's structure.

Q3: What are typical values for Circumsphere Radius?
A: The circumsphere radius depends on the pyramidal height and can vary significantly based on the specific dimensions of the polyhedron.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Great Dodecahedron due to its unique geometric properties.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with up to 10 decimal places precision, suitable for most mathematical and engineering applications.