Leg Length of Pentakis Dodecahedron given Midsphere Radius Calculator

Formula Used:

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

Leg Length (l_Leg):

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1. What is the Leg Length of Pentakis Dodecahedron?

The Leg Length of Pentakis Dodecahedron refers to the length of the equal sides of the isosceles triangular faces that make up this polyhedron. It is a crucial geometric parameter in understanding the structure and properties of the Pentakis Dodecahedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

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

Where:

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

Explanation: This formula calculates the leg length based on the given midsphere radius, incorporating the mathematical constant √5 which is fundamental to the geometry of the dodecahedron.

3. Importance of Leg Length Calculation

Details: Calculating the leg length is essential for understanding the geometric properties, surface area, volume, and other characteristics of the Pentakis Dodecahedron. It's particularly important in crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will automatically compute the corresponding leg length using the mathematical formula.

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 erected on each of its faces. It has 60 isosceles triangular faces.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the polyhedron.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact when using the formula with precise input values. The result is rounded to 6 decimal places for practical use.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. If you have measurements in other units, convert them to meters before input.

Q5: What are typical values for midsphere radius?
A: The midsphere radius depends on the specific dimensions of the Pentakis Dodecahedron. For standard configurations, it typically ranges from a few centimeters to several meters in practical applications.