Insphere Radius Of Deltoidal Hexecontahedron Given Symmetry Diagonal Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{135 + 59\sqrt{5}}{205}} \times \frac{d_{Symmetry}}{3 \times \sqrt{\frac{5 - \sqrt{5}}{20}}} \]

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 the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{135 + 59\sqrt{5}}{205}} \times \frac{d_{Symmetry}}{3 \times \sqrt{\frac{5 - \sqrt{5}}{20}}} \]

Where:

\( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
\( d_{Symmetry} \) — Symmetry Diagonal of Deltoidal Hexecontahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the insphere radius based on the symmetry diagonal measurement 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 materials science for understanding the spatial properties of polyhedra, determining packing efficiency, and analyzing the structural characteristics of crystalline formations.

4. Using the Calculator

Tips: Enter the symmetry diagonal measurement 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 Catalan solid with 60 deltoid faces, 120 edges, and 62 vertices. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: What does the symmetry diagonal represent?
A: The symmetry diagonal cuts the deltoid faces of Deltoidal Hexecontahedron into two equal halves, representing a key symmetry axis of the polyhedron.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the deltoidal hexecontahedron, assuming exact input values.

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

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, 3D modeling, and mathematical research involving polyhedral geometry.