Volume of Gyroelongated Pentagonal Pyramid Given Surface to Volume Ratio Calculator

Formula Used:

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

Volume of Gyroelongated Pentagonal Pyramid:

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1. What is Gyroelongated Pentagonal Pyramid?

A gyroelongated pentagonal pyramid is a Johnson solid constructed by attaching a pentagonal antiprism to the base of a pentagonal pyramid. It has 16 faces: 15 equilateral triangles and 1 regular pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V \) — Volume of the gyroelongated pentagonal pyramid
\( SA:V \) — Surface to volume ratio
\( \sqrt{} \) — Square root function

3. Importance of Volume Calculation

Details: Calculating the volume of complex polyhedra is important in geometry, architecture, and materials science for understanding spatial properties and structural characteristics.

4. Using the Calculator

Tips: Enter the surface to volume ratio (SA:V) in m⁻¹. The value must be greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a Johnson solid?
A: A Johnson solid is a strictly convex polyhedron with regular faces that is not uniform (not a Platonic solid, Archimedean solid, prism, or antiprism).

Q2: How many vertices and edges does a gyroelongated pentagonal pyramid have?
A: It has 11 vertices and 25 edges.

Q3: What are practical applications of this shape?
A: This geometric shape finds applications in molecular structures, architectural designs, and mathematical education.

Q4: Why is the formula so complex?
A: The complexity arises from the irregular nature of the shape and the mathematical relationships between its surface area and volume.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for gyroelongated pentagonal pyramids with known surface to volume ratios.