Insphere Radius Of Deltoidal Hexecontahedron Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{135 + 59\sqrt{5}}{205}} \times l_e \]

Insphere Radius of Deltoidal Hexecontahedron:

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1. What is the Insphere Radius of Deltoidal Hexecontahedron?

The Insphere Radius of Deltoidal Hexecontahedron is the radius of the sphere that is contained by the Deltoidal Hexecontahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{135 + 59\sqrt{5}}{205}} \times l_e \]

Where:

\( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
\( l_e \) — Long Edge of Deltoidal Hexecontahedron (m)

Explanation: This formula calculates the radius of the inscribed sphere based on the long edge length of the deltoidal hexecontahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and material science for understanding the spatial properties of polyhedrons, determining packing efficiency, and analyzing structural characteristics of crystalline forms.

4. Using the Calculator

Tips: Enter the long edge length of the deltoidal hexecontahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

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

Q2: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the largest sphere that fits inside the polyhedron, while the circumsphere radius is the radius of the smallest sphere that contains the polyhedron.

Q3: What are the practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, and the study of geometric properties of complex polyhedrons.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to the deltoidal hexecontahedron. Other polyhedrons have different formulas for calculating their insphere radii.

Q5: What precision should I use for the input values?
A: For most applications, 4-6 decimal places of precision are sufficient, though the calculator can handle higher precision if needed.