Total Surface Area of Gyroelongated Square Dipyramid given Height Calculator

Formula Used:

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

Total Surface Area (TSA):

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1. What is the Total Surface Area of Gyroelongated Square Dipyramid?

The Total Surface Area (TSA) of a Gyroelongated Square Dipyramid is the total area of all its faces combined. This geometric shape consists of triangular faces arranged in a specific configuration, and calculating its surface area involves considering the height as a key parameter.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area (m²)
\( h \) — Height of the Gyroelongated Square Dipyramid (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{3} \) — Square root of 3 (approximately 1.7321)

Explanation: The formula accounts for the geometric properties of the shape, relating the height to the total surface area through mathematical constants and operations.

3. Importance of TSA Calculation

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

4. Using the Calculator

Tips: Enter the height of the Gyroelongated Square Dipyramid in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Gyroelongated Square Dipyramid?
A: A Gyroelongated Square Dipyramid is a polyhedron formed by elongating a square dipyramid with antiprisms, resulting in a complex geometric shape with triangular faces.

Q2: Why is the formula so complex?
A: The complexity arises from the geometric relationships and trigonometric considerations involved in calculating the surface area based on the height parameter.

Q3: What units should I use for height?
A: The calculator expects height in meters, and returns surface area in square meters. Ensure consistent units for accurate results.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula is designed only for Gyroelongated Square Dipyramids. Other polyhedra require different formulas.

Q5: How accurate is the calculation?
A: The calculation is mathematically precise based on the given formula, with results rounded to 12 decimal places for clarity.