Edge Length of Elongated Pentagonal Bipyramid given Volume Calculator

Formula Used:

\[ Edge\ Length = \left( \frac{Volume}{\frac{5+\sqrt{5}}{12} + \frac{\sqrt{25+10\sqrt{5}}}{4}} \right)^{\frac{1}{3}} \]

Edge Length of Elongated Pentagonal Bipyramid:

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1. What is the Edge Length of Elongated Pentagonal Bipyramid?

The edge length of an elongated pentagonal bipyramid is a fundamental geometric measurement that represents the length of any edge of this specific polyhedron. It is derived from the volume of the shape using a mathematical formula.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ Edge\ Length = \left( \frac{Volume}{\frac{5+\sqrt{5}}{12} + \frac{\sqrt{25+10\sqrt{5}}}{4}} \right)^{\frac{1}{3}} \]

Where:

\( Volume \) — Volume of the elongated pentagonal bipyramid (m³)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the edge length by first determining the denominator which combines two geometric constants related to the pentagonal structure, then takes the cube root of the volume divided by this denominator.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the geometric properties of the elongated pentagonal bipyramid, including its surface area, spatial dimensions, and structural characteristics. This calculation is particularly important in fields such as crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the volume of the elongated pentagonal bipyramid in cubic meters. The volume must be a positive value greater than zero. The calculator will compute the corresponding edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is an elongated pentagonal bipyramid?
A: An elongated pentagonal bipyramid is a polyhedron formed by adding a pentagonal prism between two pentagonal pyramids, creating an elongated structure with specific geometric properties.

Q2: Why is the formula so complex?
A: The formula incorporates mathematical constants related to the pentagonal geometry (including the golden ratio φ) and requires square root operations to accurately represent the relationship between volume and edge length in this specific polyhedron.

Q3: What are typical values for edge length?
A: The edge length depends entirely on the volume. For larger volumes, the edge length increases, following the cube root relationship shown in the formula.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula applies only to elongated pentagonal bipyramids. Other polyhedra have different relationships between volume and edge length.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the geometric properties of elongated pentagonal bipyramids, limited only by the precision of the input values and computational floating-point arithmetic.