Long Edge Of Pentagonal Trapezohedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ Long Edge = \frac{\sqrt{5}+1}{2} \times \frac{\sqrt{\frac{25}{2} \times (5+\sqrt{5})}}{\frac{5}{12} \times (3+\sqrt{5}) \times Surface\ To\ Volume\ Ratio} \]

Long Edge Of Pentagonal Trapezohedron:

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1. What is Long Edge Of Pentagonal Trapezohedron?

The Long Edge of Pentagonal Trapezohedron is the length of the any of the longer edges of the Pentagonal Trapezohedron. It is an important geometric parameter that helps define the shape and dimensions of this particular polyhedron.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ Long Edge = \frac{\sqrt{5}+1}{2} \times \frac{\sqrt{\frac{25}{2} \times (5+\sqrt{5})}}{\frac{5}{12} \times (3+\sqrt{5}) \times Surface\ To\ Volume\ Ratio} \]

Where:

\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
Surface To Volume Ratio — The surface area to volume ratio of the pentagonal trapezohedron

Explanation: This formula calculates the long edge length based on the surface to volume ratio of the pentagonal trapezohedron, incorporating the golden ratio and geometric properties specific to this shape.

3. Importance of Long Edge Calculation

Details: Calculating the long edge of a pentagonal trapezohedron is important in geometry, crystallography, and materials science where this specific polyhedral shape occurs. It helps in understanding the dimensional relationships and geometric properties of the shape.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a pentagonal trapezohedron?
A: A pentagonal trapezohedron is a polyhedron with ten faces, each of which is a kite. It is the dual polyhedron of the pentagonal antiprism.

Q2: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the pentagonal trapezohedron. Smaller objects have higher ratios, while larger objects have lower ratios.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of a perfect pentagonal trapezohedron.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to pentagonal trapezohedra. Other polyhedra have different geometric relationships.

Q5: What units should I use for the result?
A: The result is in meters (m), consistent with the input units for surface to volume ratio (1/m).