Volume Of Deltoidal Hexecontahedron Given Insphere Radius Calculator

Formula Used:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{2 \cdot r_i}{3 \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}}} \right)^3 \]

Volume of Deltoidal Hexecontahedron:

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1. What is Volume of Deltoidal Hexecontahedron?

The volume of Deltoidal Hexecontahedron is the quantity of three dimensional space enclosed by the entire surface of Deltoidal Hexecontahedron. It is a polyhedron with deltoid (kite-shaped) faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45}{11} \cdot \sqrt{\frac{370 + 164\sqrt{5}}{25}} \cdot \left( \frac{2 \cdot r_i}{3 \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}}} \right)^3 \]

Where:

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

Explanation: This formula calculates the volume based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the deltoidal hexecontahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and various scientific applications. It helps in understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: A Deltoidal Hexecontahedron is a polyhedron with 60 deltoid (kite-shaped) faces, 120 edges, and 62 vertices.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can be contained within the polyhedron, touching all its faces.

Q3: What are the practical applications of this calculation?
A: This calculation is used in geometry research, architectural design, and in understanding the properties of complex polyhedra.

Q4: How accurate is this formula?
A: The formula is mathematically exact for a perfect deltoidal hexecontahedron with the given insphere radius.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before calculation.