Volume of Rhombicosidodecahedron given Midsphere Radius Calculator

Formula Used:

\[ V = \frac{60 + 29\sqrt{5}}{3} \times \left( \frac{2r_m}{\sqrt{10 + 4\sqrt{5}}} \right)^3 \]

Volume of Rhombicosidodecahedron:

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1. What is the Volume of Rhombicosidodecahedron?

The volume of a Rhombicosidodecahedron is the total quantity of three dimensional space enclosed by the surface of this Archimedean solid. It's a polyhedron with 20 regular triangular faces, 30 square faces, and 12 regular pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{60 + 29\sqrt{5}}{3} \times \left( \frac{2r_m}{\sqrt{10 + 4\sqrt{5}}} \right)^3 \]

Where:

\( V \) — Volume of Rhombicosidodecahedron (m³)
\( r_m \) — Midsphere Radius of Rhombicosidodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume of a Rhombicosidodecahedron based on its midsphere radius, using the mathematical relationship between these two geometric properties.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields. For Rhombicosidodecahedrons specifically, volume calculations are important in crystallography, molecular modeling, and advanced geometric studies.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombicosidodecahedron?
A: A Rhombicosidodecahedron is an Archimedean solid with 62 faces: 20 triangles, 30 squares, and 12 pentagons. It has 60 vertices and 120 edges.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the polyhedron.

Q3: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters first, then convert the result back to your desired unit.

Q4: How accurate is the calculation?
A: The calculation uses precise mathematical formulas and provides results accurate to 6 decimal places.

Q5: What are practical applications of this calculation?
A: This calculation is used in advanced geometry, architectural design, molecular modeling, and in the study of crystalline structures.