Volume of Gyroelongated Square Pyramid given Surface to Volume Ratio Calculator

Formula Used:

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

Volume of Gyroelongated Square Pyramid:

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1. What is the Volume of Gyroelongated Square Pyramid?

The Volume of Gyroelongated Square Pyramid represents the total three-dimensional space enclosed by the surface of this geometric solid. It is calculated based on the surface area to volume ratio using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Gyroelongated Square Pyramid (m³)
\( AV \) — SA:V of Gyroelongated Square Pyramid (1/m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume based on the surface area to volume ratio, incorporating mathematical constants and geometric relationships specific to the gyroelongated square pyramid.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for understanding the spatial properties of geometric solids, material requirements, structural analysis, and various engineering applications involving gyroelongated square pyramids.

4. Using the Calculator

Tips: Enter the surface area to volume ratio (SA:V) in 1/m. The value must be positive and valid 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 an elongated pyramid with rotational symmetry.

Q2: What units should be used for input?
A: The surface area to volume ratio should be entered in reciprocal meters (1/m) for consistent volume results in cubic meters (m³).

Q3: Can this calculator handle very small or large values?
A: The calculator can handle a wide range of positive values, but extremely small values may approach infinity while extremely large values approach zero in the result.

Q4: What are typical SA:V values for this shape?
A: The surface area to volume ratio depends on the specific dimensions of the pyramid, but generally falls within a range that maintains geometric feasibility.

Q5: Is this formula applicable to all pyramid shapes?
A: No, this specific formula is derived for gyroelongated square pyramids only and should not be used for other pyramid geometries.