Height of Gyroelongated Square Pyramid given Total Surface Area Calculator

Formula Used:

\[ h = \left( \sqrt{1 - \frac{1}{2 + \sqrt{2}}} + \frac{1}{\sqrt{2}} \right) \times \sqrt{\frac{S_{ATotal}}{1 + 3\sqrt{3}}} \]

Height of Gyroelongated Square Pyramid:

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1. What is the Height of Gyroelongated Square Pyramid?

The height of a gyroelongated square pyramid is the vertical distance from the apex to the base of this complex polyhedron. It represents one of the key dimensional properties of this geometric shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ h = \left( \sqrt{1 - \frac{1}{2 + \sqrt{2}}} + \frac{1}{\sqrt{2}} \right) \times \sqrt{\frac{S_{ATotal}}{1 + 3\sqrt{3}}} \]

Where:

\( h \) — Height of the gyroelongated square pyramid (m)
\( S_{ATotal} \) — Total surface area of the pyramid (m²)

Explanation: This formula derives from the geometric properties of the gyroelongated square pyramid, relating its height to its total surface area through mathematical constants and square root relationships.

3. Importance of Height Calculation

Details: Calculating the height is essential for understanding the spatial dimensions of the pyramid, which is crucial in architectural design, geometric modeling, and mathematical analysis of polyhedra.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a gyroelongated square pyramid?
A: A gyroelongated square pyramid is a Johnson solid created by attaching a square antiprism to the base of a square pyramid.

Q2: What units should I use for the input?
A: The calculator expects the total surface area in square meters (m²), and returns the height in meters (m).

Q3: Can this formula be used for other pyramid types?
A: No, this specific formula applies only to gyroelongated square pyramids due to their unique geometric properties.

Q4: What is the range of valid input values?
A: The total surface area must be greater than zero. Extremely small values may produce very small heights, while large values will produce proportionally larger heights.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the derived formula, though practical measurements of surface area may introduce some error.