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Surface to Volume Ratio of Gyroelongated Pentagonal Pyramid given Height Calculator

Formula Used:

\[ SA:V = \frac{\frac{15\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}}{\left(\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}\right) \times \frac{h}{\sqrt{1-\frac{4}{10+2\sqrt{5}}} + \sqrt{\frac{5-\sqrt{5}}{10}}}} \]

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1. What is Surface to Volume Ratio of Gyroelongated Pentagonal Pyramid?

The Surface to Volume Ratio (SA:V) of a Gyroelongated Pentagonal Pyramid is the numerical ratio of its total surface area to its volume. It's an important geometric property that describes how much surface area the shape has relative to its volume.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SA:V = \frac{\frac{15\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}}{\left(\frac{5(\sqrt{5}+1)\sqrt{\frac{10+2\sqrt{5}}{4}-1}}{3(10-2\sqrt{5})} + \frac{5+\sqrt{5}}{24}\right) \times \frac{h}{\sqrt{1-\frac{4}{10+2\sqrt{5}}} + \sqrt{\frac{5-\sqrt{5}}{10}}}} \]

Where:

Explanation: The formula calculates the ratio of total surface area to volume based on the given height of the gyroelongated pentagonal pyramid.

3. Importance of Surface to Volume Ratio Calculation

Details: Surface to volume ratio is crucial in various fields including materials science, chemistry, and engineering. It helps understand properties like heat transfer, chemical reactivity, and structural efficiency.

4. Using the Calculator

Tips: Enter the height of the gyroelongated pentagonal pyramid in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

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

Q2: What units are used for the surface to volume ratio?
A: The surface to volume ratio is measured in inverse meters (m⁻¹), as it represents area divided by volume.

Q3: Why is surface to volume ratio important?
A: It's important for understanding how the shape's properties scale with size and for applications involving surface interactions.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the standard unit. Convert other units to meters before calculation.

Q5: What is the typical range of values for this ratio?
A: The ratio depends on the height, but generally decreases as the size of the shape increases.

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