Surface to Volume Ratio of Gyroelongated Square Dipyramid given Height Calculator

Formula Used:

\[ SA:V = \frac{12\sqrt{3}}{(\sqrt{2}+\sqrt{4+3\sqrt{2}}) \times \frac{h}{\sqrt{1-\frac{1}{2+\sqrt{2}}}+\sqrt{2})} \]

Surface to Volume Ratio (SA:V):

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1. What is Surface to Volume Ratio of Gyroelongated Square Dipyramid?

The Surface to Volume Ratio (SA:V) of a Gyroelongated Square Dipyramid is a geometric measurement that represents the ratio of the total surface area to the volume of this specific polyhedron. It is an important parameter in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ SA:V = \frac{12\sqrt{3}}{(\sqrt{2}+\sqrt{4+3\sqrt{2}}) \times \frac{h}{\sqrt{1-\frac{1}{2+\sqrt{2}}}+\sqrt{2})} \]

Where:

\( h \) — Height of Gyroelongated Square Dipyramid (meters)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{3} \) — Square root of 3 (approximately 1.7321)

Explanation: This formula calculates the surface to volume ratio based on the height of the gyroelongated square dipyramid, incorporating various geometric constants and relationships specific to this polyhedron.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various fields including materials science, chemistry, and engineering. It helps in understanding properties like heat transfer, chemical reactivity, and structural efficiency of three-dimensional shapes.

4. Using the Calculator

Tips: Enter the height of the gyroelongated square dipyramid in meters. The value must be positive and greater than zero. The calculator will compute the surface to volume ratio based on the geometric properties of this specific polyhedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Gyroelongated Square Dipyramid?
A: A gyroelongated square dipyramid is a Johnson solid that can be constructed by gyroelongating a square dipyramid by inserting a square antiprism between its congruent halves.

Q2: What are typical values for surface to volume ratio?
A: The surface to volume ratio varies with the height of the polyhedron. Smaller heights generally yield higher ratios, while larger heights yield lower ratios.

Q3: What units are used in this calculation?
A: The height is input in meters (m), and the surface to volume ratio is output in reciprocal meters (m⁻¹).

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for gyroelongated square dipyramids. Other polyhedra have different geometric formulas.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the gyroelongated square dipyramid, assuming perfect geometric form.