Surface to Volume Ratio of Gyroelongated Pentagonal Pyramid given Total Surface Area Calculator

Formula Used:

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

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 a geometric measurement that compares the total surface area of the shape to its volume. It's an important parameter in various scientific and engineering applications where surface interactions are significant relative to volume.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( TSA \) — Total Surface Area of the Gyroelongated Pentagonal Pyramid (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the surface to volume ratio based on the known total surface area of the gyroelongated pentagonal pyramid, using geometric relationships specific to this polyhedron.

3. Importance of SA:V Ratio Calculation

Details: The surface to volume ratio is crucial in many fields including materials science, chemistry, and biology. It helps determine how surface properties scale with size, which affects phenomena like heat transfer, chemical reactivity, and biological processes.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding surface to volume ratio.

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: SA:V ratio indicates how much surface area is available per unit volume, which is critical for processes that occur at surfaces like catalysis, heat transfer, and diffusion.

Q3: What units does this calculator use?
A: The calculator uses square meters for surface area and returns the ratio in meters⁻¹ (inverse meters).

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to gyroelongated pentagonal pyramids as it incorporates geometric constants unique to this shape.

Q5: What if I have the volume instead of surface area?
A: This calculator specifically requires total surface area as input. A different formula would be needed if starting with volume.