Total Surface Area of Gyroelongated Pentagonal Pyramid given Height Calculator

Formula Used:

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

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 total area of all its faces, including the pentagonal base and the triangular faces. It represents the complete outer surface area of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area (m²)
\( h \) — Height of the pyramid (m)
\( \sqrt{3} \) — Square root of 3 (≈1.732)
\( \sqrt{5} \) — Square root of 5 (≈2.236)

Explanation: The formula combines geometric constants and the height parameter to calculate the total surface area of this specific polyhedral shape.

3. Importance of TSA Calculation

Details: Calculating the total surface area is important for various applications including material estimation, structural analysis, and geometric studies of complex polyhedra.

4. Using the Calculator

Tips: Enter the height of the Gyroelongated Pentagonal Pyramid in meters. The height must be a positive value greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Gyroelongated Pentagonal Pyramid?
A: It's a Johnson solid created by attaching a pentagonal pyramid to a pentagonal antiprism, resulting in a complex polyhedron with both triangular and pentagonal faces.

Q2: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the shape, which involves multiple face types and angles that require precise mathematical relationships.

Q3: What units should I use for the height?
A: The calculator uses meters, but you can use any consistent unit as long as you interpret the result in the corresponding square units.

Q4: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the shape, assuming perfect geometric construction.

Q5: Can this calculator be used for other polyhedral shapes?
A: No, this calculator is specifically designed for Gyroelongated Pentagonal Pyramids. Other polyhedral shapes require different formulas.