Volume Of Rhombicosidodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ V = \frac{60 + 29\sqrt{5}}{3} \times \left( \frac{3(30 + 5\sqrt{3}) + 3\sqrt{25 + 10\sqrt{5}}}{RA/V \times (60 + 29\sqrt{5})} \right)^3 \]

Volume of Rhombicosidodecahedron:

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1. What is the Volume of Rhombicosidodecahedron?

The Rhombicosidodecahedron is an Archimedean solid with 20 regular triangular faces, 30 square faces, and 12 regular pentagonal faces. Its volume represents the total three-dimensional space enclosed by its surface.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{60 + 29\sqrt{5}}{3} \times \left( \frac{3(30 + 5\sqrt{3}) + 3\sqrt{25 + 10\sqrt{5}}}{RA/V \times (60 + 29\sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Rhombicosidodecahedron (m³)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (≈2.23607)
\( \sqrt{3} \) — Square root of 3 (≈1.73205)

Explanation: This formula calculates the volume based on the given surface to volume ratio, using the mathematical properties of the Rhombicosidodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields where spatial measurements and properties are crucial.

4. Using the Calculator

Tips: Enter the surface to volume ratio value. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombicosidodecahedron?
A: It's one of the 13 Archimedean solids, characterized by its combination of triangular, square, and pentagonal faces in a specific arrangement.

Q2: Why is the surface to volume ratio important?
A: The surface to volume ratio is a critical parameter in many physical and chemical processes, affecting properties like heat transfer, diffusion rates, and mechanical strength.

Q3: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size and specific geometry of the solid. Smaller objects generally have higher surface to volume ratios.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Rhombicosidodecahedron only. Other polyhedra have their own unique volume formulas.

Q5: What are the practical applications of this calculation?
A: This calculation is used in crystallography, material science, architectural design, and any field dealing with complex geometric structures.