Circumsphere Radius Of Great Dodecahedron Given Ridge Length Calculator

Formula Used:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \frac{2 \times l_{Ridge}}{\sqrt{5} - 1} \]

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 in such a way that all vertices lie on the sphere's surface. It represents the distance from the center of the polyhedron to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \times \frac{2 \times l_{Ridge}}{\sqrt{5} - 1} \]

Where:

\( r_c \) — Circumsphere Radius of Great Dodecahedron (m)
\( l_{Ridge} \) — Ridge Length of Great Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the circumsphere radius based on the ridge length of the Great Dodecahedron, using mathematical constants derived from the geometric properties of the shape.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of the Great Dodecahedron, its relationship with circumscribed spheres, and for various applications in geometry, architecture, and 3D modeling.

4. Using the Calculator

Tips: Enter the ridge length of the Great Dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Dodecahedron?
A: A Great Dodecahedron is a Kepler-Poinsot polyhedron that consists of 12 pentagonal faces, with each face intersecting others in a complex pattern.

Q2: How is ridge length defined for a Great Dodecahedron?
A: Ridge Length is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Great Dodecahedron.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the ridge length. For a given ridge length, the circumsphere radius is proportionally larger based on the mathematical relationship defined by the formula.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula and calculator are designed specifically for the Great Dodecahedron geometry.

Q5: What precision does the calculator provide?
A: The calculator provides results with up to 6 decimal places precision for accurate geometric calculations.