Edge Length of Gyroelongated Square Pyramid given Volume Calculator

Formula Used:

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

Edge Length of Gyroelongated Square Pyramid (lₑ):

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1. What is the Edge Length of Gyroelongated Square Pyramid?

The edge length of a gyroelongated square pyramid is the measurement of any edge of this complex polyhedron. It is a crucial parameter that defines the size and proportions of this Johnson solid, which consists of a square pyramid attached to a square antiprism.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

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

Where:

\( l_e \) — Edge length of the gyroelongated square pyramid (m)
\( V \) — Volume of the gyroelongated square pyramid (m³)
\( \sqrt{} \) — Square root function

Explanation: This formula derives from the geometric properties of the gyroelongated square pyramid, relating its volume to the edge length through precise mathematical relationships.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from volume is essential for geometric modeling, architectural design, and understanding the spatial properties of this complex polyhedron. It helps in material estimation and structural analysis.

4. Using the Calculator

Tips: Enter the volume in cubic meters (m³). The volume must be a positive value greater than zero. The calculator will compute the corresponding edge length.

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, creating a complex polyhedron with triangular and square faces.

Q2: Why is this formula specific to this shape?
A: The formula incorporates the unique geometric constants and relationships specific to the gyroelongated square pyramid's structure.

Q3: What are typical values for edge length?
A: Edge length varies based on volume. For a volume of 1000 m³, the edge length would be approximately 10.02 meters.

Q4: Can this calculator handle very large volumes?
A: Yes, the calculator can handle large volume values, though extremely large values may be limited by computational precision.

Q5: Is this calculation reversible?
A: Yes, given the edge length, you can calculate the volume using the inverse relationship of the formula.