Midsphere Radius of Pentakis Dodecahedron Given Leg Length Calculator

Midsphere Radius of Pentakis Dodecahedron Formula:

\[ r_m = \frac{3+\sqrt{5}}{4} \times \frac{38 \times l_{Leg}}{3 \times (9+\sqrt{5})} \]

Midsphere Radius of Pentakis Dodecahedron:

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1. What is the Midsphere Radius of Pentakis Dodecahedron?

The Midsphere Radius of a Pentakis Dodecahedron is the radius of the sphere that is tangent to all the edges of the Pentakis Dodecahedron. It represents the sphere that touches every edge of the polyhedron at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3+\sqrt{5}}{4} \times \frac{38 \times l_{Leg}}{3 \times (9+\sqrt{5})} \]

Where:

\( r_m \) — Midsphere Radius of Pentakis Dodecahedron (m)
\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the leg length of the Pentakis Dodecahedron, incorporating the mathematical constant φ (phi) through the square root of 5.

3. Importance of Midsphere Radius Calculation

Details: The midsphere radius is important in geometry and 3D modeling as it helps define the spatial properties of the Pentakis Dodecahedron. It's particularly useful in crystallography, molecular modeling, and architectural design where this specific polyhedral form is employed.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that can be seen as a dodecahedron with a pyramid on each face. It has 60 isosceles triangular faces.

Q2: How is the midsphere different from the insphere?
A: The midsphere is tangent to all edges, while the insphere is tangent to all faces of the polyhedron.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the leg length. For a standard Pentakis Dodecahedron, it's typically proportional to the edge length with a factor derived from the golden ratio.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Pentakis Dodecahedron. Other polyhedra have different formulas for calculating their midsphere radii.

Q5: What practical applications does this calculation have?
A: This calculation is used in various fields including crystallography (for certain crystal structures), molecular modeling (for fullerene structures), and architectural design (for geodesic domes and other structures).