Insphere Radius of Deltoidal Icositetrahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_i = \sqrt{\frac{22 + 15\sqrt{2}}{34}} \times \frac{6}{SA:V} \times \sqrt{\frac{61 + 38\sqrt{2}}{292 + 206\sqrt{2}}} \]

Insphere Radius (r_i):

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1. What is 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 represents the distance from the center of the polyhedron to the center of any of its deltoidal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \sqrt{\frac{22 + 15\sqrt{2}}{34}} \times \frac{6}{SA:V} \times \sqrt{\frac{61 + 38\sqrt{2}}{292 + 206\sqrt{2}}} \]

Where:

\( r_i \) — Insphere radius (m)
\( SA:V \) — Surface to Volume Ratio (1/m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula relates the insphere radius to the surface-to-volume ratio through a complex mathematical relationship involving square roots and rational expressions.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the packing efficiency, structural properties, and spatial relationships within the deltoidal icositetrahedron shape.

4. Using the Calculator

Tips: Enter the surface-to-volume ratio in 1/m. 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 deltoidal (kite-shaped) faces, 26 vertices, and 48 edges.

Q2: How is surface-to-volume ratio defined?
A: Surface-to-volume ratio is the total surface area divided by the total volume of the polyhedron, measured in 1/m.

Q3: What are typical values for insphere radius?
A: The insphere radius depends on the size of the polyhedron and its surface-to-volume ratio, typically ranging from fractions of a meter to several meters.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal icositetrahedron due to its unique geometric properties.

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, architectural design, and the study of geometric properties of complex polyhedra.