Volume Of Icosidodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ V = \frac{45 + 17\sqrt{5}}{6} \times \left( \frac{6 \times (5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}})}{RA/V \times (45 + 17\sqrt{5})} \right)^3 \]

Volume of Icosidodecahedron:

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1. What is the Volume of Icosidodecahedron?

The volume of an icosidodecahedron represents the total three-dimensional space enclosed by its surface. It is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 60 edges, and 30 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45 + 17\sqrt{5}}{6} \times \left( \frac{6 \times (5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}})}{RA/V \times (45 + 17\sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Icosidodecahedron (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 icosidodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields. It helps in understanding spatial properties, material requirements, and structural characteristics.

4. Using the Calculator

Tips: Enter the surface to volume ratio value in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is an icosidodecahedron?
A: An icosidodecahedron is an Archimedean solid with 32 faces (20 equilateral triangles and 12 regular pentagons), 60 edges, and 30 vertices.

Q2: What are typical surface to volume ratio values?
A: The surface to volume ratio depends on the size of the icosidodecahedron. Smaller solids have higher ratios, while larger ones have lower ratios.

Q3: Can this calculator handle different units?
A: The calculator uses meters as the base unit. Ensure consistent units when inputting values and interpreting results.

Q4: What is the significance of the mathematical constants in the formula?
A: The constants √5 and √3 are fundamental mathematical constants that appear in the geometric properties of the icosidodecahedron.

Q5: How accurate is the calculation?
A: The calculation provides high mathematical accuracy based on the precise geometric formula for the icosidodecahedron.