Volume Of Deltoidal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{20 \cdot r_m}{3 \cdot (5 + 3\sqrt{5})} \right)^3 \]

Volume (V):

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1. What is the Volume of Deltoidal Hexecontahedron?

The volume of a Deltoidal Hexecontahedron is the quantity of three dimensional space enclosed by the entire surface of this polyhedron. It's a Catalan solid with 60 deltoid faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{20 \cdot r_m}{3 \cdot (5 + 3\sqrt{5})} \right)^3 \]

Where:

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

Explanation: This formula calculates the volume based on the midsphere radius, incorporating the mathematical constant √5 which is characteristic of many geometric calculations involving pentagonal symmetry.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in various fields including mathematics, engineering, architecture, and material science. For the Deltoidal Hexecontahedron, volume calculation helps in understanding its spatial properties and applications in design and modeling.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: It's a Catalan solid with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices. It's the dual polyhedron of the rhombicosidodecahedron.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that touches all the edges of the polyhedron. For a Deltoidal Hexecontahedron, it's the sphere tangent to all its edges.

Q3: Why does the formula contain √5?
A: The presence of √5 in the formula reflects the pentagonal symmetry inherent in the Deltoidal Hexecontahedron, which is related to the golden ratio φ = (1+√5)/2.

Q4: What are practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, crystallography, and in the study of polyhedral structures in mathematics and materials science.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact when using the formula. The accuracy of the result depends on the precision of the input value and the computational precision of the calculator.