Midsphere Radius of Pentakis Dodecahedron given Volume Calculator

Formula Used:

\[ r_m = \frac{3 + \sqrt{5}}{4} \times \left( \frac{76 \times V}{15 \times (23 + 11 \times \sqrt{5})} \right)^{\frac{1}{3}} \]

Midsphere Radius of Pentakis Dodecahedron (rₘ):

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1. What is the Midsphere Radius of Pentakis Dodecahedron?

The Midsphere Radius of Pentakis Dodecahedron is the radius of the sphere for which all the edges of the Pentakis Dodecahedron become a tangent line on that sphere. It is an important geometric property of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3 + \sqrt{5}}{4} \times \left( \frac{76 \times V}{15 \times (23 + 11 \times \sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( r_m \) — Midsphere Radius of Pentakis Dodecahedron (m)
\( V \) — Volume of Pentakis Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the volume of the Pentakis Dodecahedron, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties and relationships of the Pentakis Dodecahedron. It helps in various applications including crystallography, architectural design, and mathematical research.

4. Using the Calculator

Tips: Enter the volume of the Pentakis Dodecahedron in cubic meters. The value must be positive and valid. 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 isosceles triangular faces.

Q2: What are the applications of this calculation?
A: This calculation is used in geometry, 3D modeling, architectural design, and in the study of polyhedral structures.

Q3: What units should be used for volume?
A: Volume should be entered in cubic meters (m³) for consistent results with the radius in meters.

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

Q5: What is the significance of the constants in the formula?
A: The constants (3, 4, 76, 15, 23, 11) are derived from the geometric properties and mathematical relationships specific to the Pentakis Dodecahedron.