Height of Pentagonal Bipyramid given Volume Calculator

Formula Used:

\[ h = 2 \times \sqrt{\frac{5 - \sqrt{5}}{10}} \times \left( \frac{12 \times V}{5 + \sqrt{5}} \right)^{\frac{1}{3}} \]

Height of Pentagonal Bipyramid (h):

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1. What is the Height of Pentagonal Bipyramid?

The height of a pentagonal bipyramid is the vertical distance between the two apex vertices of the bipyramid. It is an important geometric measurement that helps define the overall dimensions and proportions of this polyhedral structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = 2 \times \sqrt{\frac{5 - \sqrt{5}}{10}} \times \left( \frac{12 \times V}{5 + \sqrt{5}} \right)^{\frac{1}{3}} \]

Where:

\( h \) — Height of Pentagonal Bipyramid (m)
\( V \) — Volume of Pentagonal Bipyramid (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the geometric relationships between the height, volume, and base dimensions of a regular pentagonal bipyramid.

3. Importance of Height Calculation

Details: Calculating the height is essential for understanding the spatial dimensions of the bipyramid, which is important in various applications including crystallography, molecular modeling, and architectural design.

4. Using the Calculator

Tips: Enter the volume of the pentagonal bipyramid in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a pentagonal bipyramid?
A: A pentagonal bipyramid is a polyhedron formed by two pentagonal pyramids sharing a common pentagonal base. It has 7 vertices and 10 faces.

Q2: What are the applications of pentagonal bipyramids?
A: They are used in chemistry (molecular structures), crystallography, and as geometric models in various scientific and engineering fields.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular pentagonal bipyramid with equal edge lengths.

Q4: Can this formula be used for irregular bipyramids?
A: No, this formula applies only to regular pentagonal bipyramids where all edges are equal and the base is a regular pentagon.

Q5: What units should I use for volume?
A: The calculator expects volume in cubic meters, but you can use any consistent unit system as long as the height will be in the corresponding linear units.