Total Surface Area of Gyroelongated Pentagonal Pyramid given Volume Calculator

Formula Used:

\[ TSA = \frac{15\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4} \times \left( \frac{V}{\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}} \right)^{\frac{2}{3}} \]

Total Surface Area (TSA):

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1. What is the Total Surface Area of Gyroelongated Pentagonal Pyramid?

The Total Surface Area (TSA) of a Gyroelongated Pentagonal Pyramid is the sum of the areas of all its faces. It includes the area of the pentagonal base and the triangular faces that form the pyramid structure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4} \times \left( \frac{V}{\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}} \right)^{\frac{2}{3}} \]

Where:

\( TSA \) — Total Surface Area (m²)
\( V \) — Volume of the Gyroelongated Pentagonal Pyramid (m³)

Explanation: The formula calculates the total surface area based on the given volume, using geometric relationships specific to this polyhedral shape.

3. Importance of TSA Calculation

Details: Calculating the total surface area is important for various applications including material estimation, structural analysis, and understanding the geometric properties of this complex polyhedron.

4. Using the Calculator

Tips: Enter the volume of the Gyroelongated Pentagonal Pyramid in cubic meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Gyroelongated Pentagonal Pyramid?
A: A Gyroelongated Pentagonal Pyramid is a Johnson solid formed by attaching a pentagonal pyramid to a pentagonal base and rotating it.

Q2: What units should I use for volume input?
A: The calculator expects volume input in cubic meters (m³). Make sure to convert from other units if necessary.

Q3: How accurate is this calculation?
A: The calculation uses precise mathematical formulas and provides results accurate to 6 decimal places.

Q4: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, as long as they are positive numbers.

Q5: What are the practical applications of this calculation?
A: This calculation is useful in architecture, engineering, 3D modeling, and mathematical research involving polyhedral geometry.