Total Surface Area of Elongated Pentagonal Bipyramid given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \left(\frac{5\sqrt{3}}{2}+5\right) \times \left(\frac{\frac{5\sqrt{3}}{2}+5}{\left(\frac{5+\sqrt{5}}{12}+\frac{\sqrt{25+10\sqrt{5}}}{4}\right) \times SA:V}\right)^2 \]

Total Surface Area (TSA):

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1. What is the Elongated Pentagonal Bipyramid?

The Elongated Pentagonal Bipyramid is a polyhedron formed by adding two pentagonal pyramids to a pentagonal prism. It has 12 faces (10 equilateral triangles and 2 pentagons), 20 edges, and 12 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \left(\frac{5\sqrt{3}}{2}+5\right) \times \left(\frac{\frac{5\sqrt{3}}{2}+5}{\left(\frac{5+\sqrt{5}}{12}+\frac{\sqrt{25+10\sqrt{5}}}{4}\right) \times SA:V}\right)^2 \]

Where:

\( TSA \) — Total Surface Area (m²)
\( SA:V \) — Surface to Volume Ratio (1/m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area based on the given surface to volume ratio, using geometric properties of the elongated pentagonal bipyramid.

3. Importance of Surface Area Calculation

Details: Calculating surface area is important in various applications including material science, heat transfer calculations, and structural analysis where surface properties affect performance.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the typical range for SA:V values?
A: SA:V values depend on the size of the polyhedron. Smaller polyhedra typically have higher SA:V ratios.

Q2: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the elongated pentagonal bipyramid geometry.

Q3: What units should I use for the input?
A: The calculator expects SA:V in 1/m (inverse meters) and returns surface area in square meters.

Q4: How accurate is the calculation?
A: The calculation uses precise mathematical formulas and provides results accurate to 6 decimal places.

Q5: What if I get an error in calculation?
A: Ensure you've entered a valid positive number for SA:V ratio and that the value is within reasonable bounds.