Edge Length of Gyroelongated Pentagonal Pyramid given Volume Calculator

Formula Used:

\[ l_e = \left( \frac{V}{\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}} \right)^{\frac{1}{3}} \]

Edge Length of Gyroelongated Pentagonal Pyramid:

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1. What is the Edge Length of Gyroelongated Pentagonal Pyramid?

The edge length of a gyroelongated pentagonal pyramid is the measurement of any edge of this complex polyhedron. It is a fundamental geometric property used in various mathematical and engineering calculations.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \left( \frac{V}{\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}} \right)^{\frac{1}{3}} \]

Where:

\( l_e \) — Edge Length of Gyroelongated Pentagonal Pyramid (m)
\( V \) — Volume of Gyroelongated Pentagonal Pyramid (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives the edge length from the volume of the gyroelongated pentagonal pyramid using geometric relationships and mathematical constants.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the geometric properties of the polyhedron, construction applications, and mathematical modeling of complex shapes.

4. Using the Calculator

Tips: Enter the volume of the gyroelongated pentagonal pyramid in cubic meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a gyroelongated pentagonal pyramid?
A: A gyroelongated pentagonal pyramid is a Johnson solid formed by attaching a pentagonal antiprism to the base of a pentagonal pyramid.

Q2: What are typical edge length values?
A: Edge length values vary depending on the volume, but typically range from small fractions to several meters in practical applications.

Q3: What units should be used for volume input?
A: The calculator expects volume input in cubic meters (m³) for consistent results.

Q4: Are there limitations to this calculation?
A: The formula assumes a perfect geometric shape and may not account for manufacturing tolerances or material properties in real-world applications.

Q5: Can this calculator handle very large or small volumes?
A: Yes, the calculator can handle a wide range of volume values, but extremely large or small values may be limited by computational precision.