Volume of Pentagonal Bipyramid given Surface to Volume Ratio Calculator

Formula Used:

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

Volume of Pentagonal Bipyramid:

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1. What is the Volume of Pentagonal Bipyramid?

The volume of a pentagonal bipyramid refers to the total three-dimensional space enclosed by the surface of this geometric shape. A pentagonal bipyramid consists of two pentagonal pyramids joined base-to-base.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Pentagonal Bipyramid (m³)
\( \frac{A}{V} \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the volume of a pentagonal bipyramid based on its surface to volume ratio, using geometric relationships specific to this polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in various fields including architecture, engineering, materials science, and 3D modeling. Understanding volume helps in determining capacity, material requirements, and structural properties.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in the input field. The value must be a positive number greater than zero. The calculator will compute the corresponding volume of the pentagonal bipyramid.

5. Frequently Asked Questions (FAQ)

Q1: What is a pentagonal bipyramid?
A: A pentagonal bipyramid is a polyhedron formed by two pentagonal pyramids joined base-to-base, resulting in a shape with 7 vertices, 15 edges, and 10 triangular faces.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is crucial in many applications including heat transfer, chemical reactions, and biological systems where surface area relative to volume affects various physical and chemical processes.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size and proportions of the specific pentagonal bipyramid. Smaller objects generally have higher surface to volume ratios.

Q4: Can this calculator handle different units?
A: The calculator uses consistent units (meters for length, cubic meters for volume). Ensure all inputs use compatible units for accurate results.

Q5: What if I get an unexpected result?
A: Double-check your input value and ensure it's a positive number. The formula is specifically designed for pentagonal bipyramids and may not be applicable to other shapes.