Surface to Volume Ratio of Gyroelongated Pentagonal Pyramid given Volume Calculator

Formula Used:

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

Surface to Volume Ratio (SA:V):

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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 following formula:

Where:

\( V \) — Volume of the gyroelongated pentagonal pyramid (m³)
\( \sqrt{} \) — Square root function

Explanation: This complex formula calculates the surface to volume ratio based on the volume input, using geometric properties specific to the gyroelongated pentagonal pyramid shape.

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 understand properties like heat transfer, reaction rates, and structural efficiency in geometric shapes.

4. Using the Calculator

Tips: Enter the volume of the gyroelongated pentagonal 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 pentagonal pyramid?
A: A gyroelongated pentagonal pyramid is a Johnson solid created by attaching a pentagonal antiprism to the base of a pentagonal pyramid.

Q2: Why is surface to volume ratio important?
A: It indicates how much surface area is available per unit volume, which affects properties like heat dissipation, chemical reactivity, and structural stability.

Q3: What units are used in this calculation?
A: Volume is in cubic meters (m³) and surface to volume ratio is in reciprocal meters (m⁻¹).

Q4: 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 affect computational precision.

Q5: Is this formula specific to gyroelongated pentagonal pyramids?
A: Yes, this formula is specifically derived for calculating the surface to volume ratio of gyroelongated pentagonal pyramids given their volume.