Leg Length of Pentakis Dodecahedron given Total Surface Area Calculator

Formula Used:

\[ l_{Leg} = \frac{3}{38} \times (9 + \sqrt{5}) \times \sqrt{\frac{19 \times TSA}{15 \times \sqrt{413 + 162 \times \sqrt{5}}}}} \]

Leg Length (lLeg):

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1. What is the Leg Length of Pentakis Dodecahedron?

The Leg Length of Pentakis Dodecahedron is the length of the equal sides of the isosceles triangular face of a Pentakis Dodecahedron. It is a key geometric parameter in this polyhedron structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Leg} = \frac{3}{38} \times (9 + \sqrt{5}) \times \sqrt{\frac{19 \times TSA}{15 \times \sqrt{413 + 162 \times \sqrt{5}}}}} \]

Where:

\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( TSA \) — Total Surface Area of Pentakis Dodecahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the leg length based on the total surface area of the Pentakis Dodecahedron, incorporating mathematical constants and square roots.

3. Importance of Leg Length Calculation

Details: Calculating the leg length is important for geometric analysis, architectural design, and understanding the properties of the Pentakis Dodecahedron structure.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid.

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: Why is this formula so complex?
A: The formula incorporates mathematical constants and relationships specific to the geometry of the Pentakis Dodecahedron, resulting in its complexity.

Q3: What are typical values for leg length?
A: The leg length varies based on the total surface area. For a given surface area, the calculator provides the corresponding leg length.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for the Pentakis Dodecahedron and its leg length calculation.

Q5: How accurate are the results?
A: The results are mathematically precise based on the input values and the formula, with rounding to 10 decimal places for clarity.