Total Surface Area of Gyroelongated Square Dipyramid given Volume Calculator

Formula Used:

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

Total Surface Area (TSA):

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1. What is the Gyroelongated Square Dipyramid?

The Gyroelongated Square Dipyramid is a Johnson solid formed by gyroelongating a square dipyramid. It consists of 16 equilateral triangles and has a complex geometric structure with rotational symmetry.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area (m²)
\( V \) — Volume of the Gyroelongated Square Dipyramid (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives the total surface area from the volume by utilizing the geometric properties and mathematical relationships specific to this Johnson solid.

3. Importance of TSA Calculation

Details: Calculating the total surface area is essential for understanding the material requirements, heat transfer properties, and other physical characteristics of this geometric shape in practical applications.

4. Using the Calculator

Tips: Enter the volume in cubic meters. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Johnson solid?
A: Johnson solids are convex polyhedra with regular faces that are not uniform (unlike Platonic or Archimedean solids).

Q2: How many faces does a Gyroelongated Square Dipyramid have?
A: It has 16 faces, all of which are equilateral triangles.

Q3: What are practical applications of this shape?
A: While primarily mathematical, such shapes find applications in architecture, molecular structures, and geometric art.

Q4: Why is the formula so complex?
A: The complexity arises from the intricate geometric relationships between volume and surface area in this specific polyhedron.

Q5: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Gyroelongated Square Dipyramid due to its unique geometric properties.