Total Surface Area of Rhombicosidodecahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ TSA = \left(30 + 5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right) \times \left(\frac{3 \times \left(30 + 5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right)}{RA/V \times \left(60 + 29\sqrt{5}\right)}\right)^2 \]

Total Surface Area (TSA):

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1. What is the Total Surface Area of Rhombicosidodecahedron?

The Total Surface Area of a Rhombicosidodecahedron is the total quantity of plane enclosed by the entire surface of this Archimedean solid. It consists of 20 regular triangular faces, 30 square faces, and 12 regular pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \left(30 + 5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right) \times \left(\frac{3 \times \left(30 + 5\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right)}{RA/V \times \left(60 + 29\sqrt{5}\right)}\right)^2 \]

Where:

\( TSA \) — Total Surface Area
\( RA/V \) — Surface to Volume Ratio
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula derives the total surface area from the surface to volume ratio using the geometric properties of the rhombicosidodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material science, architectural design, and understanding the geometric properties of this complex polyhedron.

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: A rhombicosidodecahedron is an Archimedean solid with 20 triangular faces, 30 square faces, and 12 pentagonal faces, totaling 62 faces.

Q2: What units should I use for surface to volume ratio?
A: The surface to volume ratio should be in reciprocal meters (1/m) to maintain dimensional consistency.

Q3: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of values, but extremely small values may approach computational limits.

Q4: What are typical surface to volume ratio values for this shape?
A: The surface to volume ratio depends on the size of the polyhedron, with smaller objects having higher ratios.

Q5: Is this formula exact or approximate?
A: This is an exact mathematical formula derived from the geometric properties of the rhombicosidodecahedron.