Height of Gyroelongated Square Pyramid given Volume Calculator

Formula Used:

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

Height of Gyroelongated Square Pyramid (h):

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1. What is the Height of Gyroelongated Square Pyramid?

The Height of Gyroelongated Square Pyramid is the vertical distance from the highest point to the lowest point of the Gyroelongated Square Pyramid. It is an important geometric measurement in 3D space analysis.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

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

Where:

\( h \) — Height of Gyroelongated Square Pyramid (m)
\( V \) — Volume of Gyroelongated Square Pyramid (m³)

Explanation: This formula calculates the height based on the volume of the gyroelongated square pyramid, incorporating square roots and cube roots to account for the complex geometry.

3. Importance of Height Calculation

Details: Calculating the height of a gyroelongated square pyramid is essential for architectural design, structural analysis, and geometric modeling applications where precise dimensional measurements are required.

4. Using the Calculator

Tips: Enter the volume of the gyroelongated square pyramid in cubic 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 constructed by attaching a square pyramid to a square antiprism, creating a complex polyhedral structure.

Q2: Why is the formula so complex?
A: The complexity arises from the irregular geometry of the gyroelongated square pyramid, which requires advanced mathematical operations to relate volume to height.

Q3: What units should I use for volume?
A: The calculator expects volume in cubic meters (m³), but you can use any consistent unit system as long as you interpret the height result in the same unit length.

Q4: Are there limitations to this calculation?
A: This formula assumes a perfect geometric shape and may not account for manufacturing tolerances or material deformations in real-world applications.

Q5: Can this calculator handle very large or very small volumes?
A: The calculator can handle a wide range of volume values, but extremely large or small values may be limited by PHP's floating-point precision.