Insphere Radius of Deltoidal Icositetrahedron given Total Surface Area Calculator

Formula Used:

\[ r_i = \sqrt{\frac{22+15\sqrt{2}}{34}} \times \sqrt{\frac{7 \times TSA}{12 \times \sqrt{61+38\sqrt{2}}}} \]

Insphere Radius (rᵢ):

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1. What is the Insphere Radius of Deltoidal Icositetrahedron?

The Insphere Radius of a Deltoidal Icositetrahedron is the radius of the largest sphere that can be inscribed within the polyhedron such that it touches all the faces tangentially. It provides insight into the internal geometry and spatial properties of this complex 3D shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \sqrt{\frac{22+15\sqrt{2}}{34}} \times \sqrt{\frac{7 \times TSA}{12 \times \sqrt{61+38\sqrt{2}}}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( TSA \) — Total Surface Area (m²)

Explanation: This formula derives from the geometric properties of the Deltoidal Icositetrahedron, relating the insphere radius to the total surface area through specific mathematical constants.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is essential for understanding the internal packing efficiency, material distribution, and geometric optimization of the Deltoidal Icositetrahedron in various mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding insphere radius.

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: How accurate is this formula?
A: The formula is mathematically exact for a perfect Deltoidal Icositetrahedron, derived from its geometric properties.

Q3: Can this calculator handle different units?
A: The calculator uses square meters for surface area and meters for radius. Convert other units to these before calculation.

Q4: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a unit surface area, the radius follows the proportional relationship defined by the formula.

Q5: Are there limitations to this calculation?
A: This calculation assumes a perfect geometric shape. Real-world approximations may have slight variations due to manufacturing tolerances.