Insphere Radius of Deltoidal Icositetrahedron given NonSymmetry Diagonal Calculator

Formula Used:

\[ r_i = \sqrt{\frac{22 + 15\sqrt{2}}{34}} \times \frac{2 \times d_{Non\ Symmetry}}{\sqrt{4 + 2\sqrt{2}}} \]

Insphere Radius of Deltoidal Icositetrahedron:

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1. What is the Insphere Radius of Deltoidal Icositetrahedron?

The Insphere Radius of Deltoidal Icositetrahedron is the radius of the sphere that is contained by the Deltoidal Icositetrahedron 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:

\[ r_i = \sqrt{\frac{22 + 15\sqrt{2}}{34}} \times \frac{2 \times d_{Non\ Symmetry}}{\sqrt{4 + 2\sqrt{2}}} \]

Where:

\( r_i \) — Insphere Radius of Deltoidal Icositetrahedron (m)
\( d_{Non\ Symmetry} \) — NonSymmetry Diagonal of Deltoidal Icositetrahedron (m)

Explanation: This formula calculates the insphere radius based on the non-symmetry diagonal measurement of the deltoidal icositetrahedron, incorporating mathematical constants and square root functions.

3. Importance of Insphere Radius Calculation

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

4. Using the Calculator

Tips: Enter the NonSymmetry 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 Icositetrahedron?
A: A Deltoidal Icositetrahedron is a Catalan solid with 24 deltoid (kite-shaped) faces, 26 vertices, and 48 edges.

Q2: What is the difference between symmetry and non-symmetry diagonals?
A: Non-symmetry diagonals are those that divide the deltoid faces into two isosceles triangles, while symmetry diagonals follow the symmetry axes of the polyhedron.

Q3: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a unit deltoidal icositetrahedron, the insphere radius is approximately 0.919 m.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the deltoidal icositetrahedron and cannot be directly applied to other polyhedral shapes.

Q5: What are the practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, and mathematical modeling where precise geometric measurements of polyhedral structures are required.