Edge Length of Gyroelongated Pentagonal Pyramid given Surface to Volume Ratio Calculator

Formula Used:

\[ l_e = \frac{\frac{15\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}}{\left(\frac{5(\sqrt{5} + 1)\sqrt{\frac{10 + 2\sqrt{5}}{4} - 1}}{3(10 - 2\sqrt{5})} + \frac{5 + \sqrt{5}}{24}\right) \times \text{SA:V}} \]

Edge Length:

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1. What is the Gyroelongated Pentagonal Pyramid?

The Gyroelongated Pentagonal Pyramid is a Johnson solid formed by attaching a pentagonal antiprism to a pentagonal pyramid. It has 16 faces: 15 equilateral triangles and 1 regular pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( l_e \) — Edge length of the pyramid
SA:V — Surface to volume ratio of the pyramid
Mathematical constants derived from the geometry of the solid

Explanation: The formula calculates the edge length based on the known surface to volume ratio of the gyroelongated pentagonal pyramid.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the surface to volume ratio is important in materials science, crystallography, and architectural design where specific surface area properties are required.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Johnson solid?
A: Johnson solids are convex polyhedra with regular faces that are not uniform (Platonic or Archimedean solids).

Q2: How many vertices and edges does a gyroelongated pentagonal pyramid have?
A: It has 11 vertices and 25 edges.

Q3: What are practical applications of this calculation?
A: This calculation is useful in nanotechnology, material design, and architectural structures where specific surface area properties are important.

Q4: Are there limitations to this formula?
A: The formula assumes a perfect geometric shape and may not account for manufacturing tolerances or material imperfections.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the gyroelongated pentagonal pyramid geometry.