Volume of Rhombic Triacontahedron given Midsphere Radius Calculator

Formula Used:

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

Volume of Rhombic Triacontahedron:

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1. What is the Volume of Rhombic Triacontahedron?

The Volume of Rhombic Triacontahedron is the quantity of three dimensional space enclosed by the entire surface of Rhombic Triacontahedron. It is a Catalan solid with 30 rhombic faces and has interesting mathematical properties.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Rhombic Triacontahedron (m³)
\( r_m \) — Midsphere Radius of Rhombic Triacontahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume of a rhombic triacontahedron based on its midsphere radius, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For the rhombic triacontahedron specifically, volume calculations are important in crystallography, material science, and mathematical research on polyhedra.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive (radius > 0). The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a rhombic triacontahedron?
A: A rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is one of the Catalan solids and is the dual polyhedron of the icosidodecahedron.

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: Are there other ways to calculate the volume?
A: Yes, the volume can also be calculated using edge length or other parameters, but this calculator specifically uses the midsphere radius.

Q4: What are the practical applications of this calculation?
A: This calculation is used in mathematical research, geometric modeling, crystal structure analysis, and architectural design involving complex polyhedra.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula, providing precise volume measurements for the specified midsphere radius.