Total Surface Area of Gyroelongated Square Pyramid given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = (1+3\sqrt{3}) \times \left( \frac{1+3\sqrt{3}}{\left( \frac{\sqrt{4+3\sqrt{2}} + \frac{1}{\sqrt{2}}}{3} \times SA:V \right)} \right)^2 \]

Total Surface Area (TSA):

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1. What is the Gyroelongated Square Pyramid?

The Gyroelongated Square Pyramid is a Johnson solid constructed by attaching a square pyramid to a square antiprism. It has 13 faces: 12 equilateral triangles and 1 square.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = (1+3\sqrt{3}) \times \left( \frac{1+3\sqrt{3}}{\left( \frac{\sqrt{4+3\sqrt{2}} + \frac{1}{\sqrt{2}}}{3} \times SA:V \right)} \right)^2 \]

Where:

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

Explanation: This formula calculates the total surface area of a gyroelongated square pyramid based on its surface to volume ratio, incorporating geometric constants specific to this polyhedron.

3. Importance of Surface Area Calculation

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

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a gyroelongated square pyramid?
A: It's a Johnson solid (J10) formed by attaching a square pyramid to a square antiprism, resulting in a polyhedron with 13 faces.

Q2: What are the practical applications of this calculation?
A: This calculation is useful in architecture, material science, and 3D modeling where precise surface area measurements are required.

Q3: How accurate is this formula?
A: The formula is mathematically exact for perfect gyroelongated square pyramids and provides precise surface area calculations.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the base unit. For other units, appropriate conversion factors must be applied to the input and output values.

Q5: What is the typical range of SA:V values for this shape?
A: The surface to volume ratio varies with size, but generally falls within a specific range characteristic of this geometric shape.