Circumsphere Radius of Small Stellated Dodecahedron given Pyramidal Height Calculator

Formula Used:

\[ r_c = \frac{\sqrt{50+22\sqrt{5}}}{4} \times \frac{5h_{Pyramid}}{\sqrt{25+10\sqrt{5}}} \]

Circumsphere Radius of Small Stellated Dodecahedron:

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1. What is Circumsphere Radius of Small Stellated Dodecahedron?

The Circumsphere Radius of a Small Stellated Dodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It is a fundamental geometric property of this complex polyhedral shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_c = \frac{\sqrt{50+22\sqrt{5}}}{4} \times \frac{5h_{Pyramid}}{\sqrt{25+10\sqrt{5}}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( h_{Pyramid} \) — Pyramidal Height of Small Stellated Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula relates the circumsphere radius to the pyramidal height through a complex mathematical relationship involving square roots and the golden ratio constant (√5).

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and geometric properties of the Small Stellated Dodecahedron, which is important in mathematical modeling, architectural design, and geometric analysis.

4. Using the Calculator

Tips: Enter the pyramidal height value 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 Small Stellated Dodecahedron?
A: It is a Kepler-Poinsot polyhedron that consists of 12 pentagrammic faces with the Schläfli symbol {5/2,5}. It is one of four regular star polyhedra.

Q2: How is Pyramidal Height defined for this polyhedron?
A: Pyramidal Height refers to the height of any of the inwards directed tetrahedral pyramids that form the stellated structure of the dodecahedron.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the specific dimensions of the polyhedron. For standard units, it typically ranges from a few centimeters to several meters depending on the scale.

Q4: Are there practical applications of this calculation?
A: Yes, this calculation is used in mathematical research, architectural design of complex structures, and in understanding the geometric properties of star polyhedra.

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