Midsphere Radius Of Pentakis Dodecahedron Given Insphere Radius Calculator

Formula Used:

\[ r_m = \frac{3 + \sqrt{5}}{4} \times \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \]

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 is an important geometric property that helps in understanding the spatial characteristics of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3 + \sqrt{5}}{4} \times \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \]

Where:

\( r_m \) — Midsphere Radius of Pentakis Dodecahedron (m)
\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes a mathematical relationship between the midsphere radius and the insphere radius of a Pentakis Dodecahedron, incorporating the golden ratio through the square root of 5.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is essential for geometric analysis, 3D modeling, and understanding the symmetrical properties of the Pentakis Dodecahedron in various mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the insphere radius value 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 is the dual of the truncated icosahedron. It has 60 faces, 90 edges, and 32 vertices.

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

Q3: What are typical values for these radii?
A: The values depend on the specific dimensions of the Pentakis Dodecahedron. Both radii are proportional to the overall size of the polyhedron.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Pentakis Dodecahedron. Other polyhedra have different geometric relationships.

Q5: What practical applications does this calculation have?
A: This calculation is useful in crystallography, architecture, and computer graphics where precise geometric modeling of Pentakis Dodecahedra is required.