Volume Of Deltoidal Hexecontahedron Given Short Edge Calculator

Formula Used:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{22 \cdot l_{short}}{3 \cdot (7 - \sqrt{5})} \right)^3 \]

Volume of Deltoidal Hexecontahedron:

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1. What is 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 is a Catalan solid with 60 deltoidal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{22 \cdot l_{short}}{3 \cdot (7 - \sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Deltoidal Hexecontahedron (m³)
\( l_{short} \) — Short Edge of Deltoidal Hexecontahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume based on the short edge length of the deltoidal faces, incorporating 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 Deltoidal Hexecontahedron, understanding its volume helps in material estimation, structural analysis, and mathematical modeling of complex polyhedra.

4. Using the Calculator

Tips: Enter the short edge length 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 Deltoidal Hexecontahedron?
A: It's a Catalan solid with 60 deltoidal (kite-shaped) faces, 120 edges, and 62 vertices. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: What are the typical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, mathematical research, and educational contexts to understand polyhedral properties.

Q3: How accurate is this formula?
A: The formula is mathematically exact for a perfect Deltoidal Hexecontahedron. The calculator provides results with high precision based on the input values.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, or convert the result from cubic meters to your desired volume unit.

Q5: What is the range of valid input values?
A: The short edge length must be a positive real number. Extremely large values may cause computational limitations in some browsers.