Short Edge of Deltoidal Icositetrahedron given Volume Calculator

Formula Used:

\[ \text{Short Edge} = \frac{4+\sqrt{2}}{7} \times \left( \frac{7 \times V}{2 \times \sqrt{292 + (206 \times \sqrt{2})}} \right)^{\frac{1}{3}} \]

Short Edge of Deltoidal Icositetrahedron:

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1. What is the Short Edge of Deltoidal Icositetrahedron?

The Short Edge of Deltoidal Icositetrahedron is the length of the shortest edge of the identical deltoidal faces that make up the Deltoidal Icositetrahedron, a Catalan solid with 24 congruent deltoid faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{4+\sqrt{2}}{7} \times \left( \frac{7 \times V}{2 \times \sqrt{292 + (206 \times \sqrt{2})}} \right)^{\frac{1}{3}} \]

Where:

\( V \) — Volume of Deltoidal Icositetrahedron (m³)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: The formula calculates the shortest edge length based on the given volume of the Deltoidal Icositetrahedron, incorporating geometric constants specific to this polyhedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for understanding the geometry of the Deltoidal Icositetrahedron, including its surface area, other edge lengths, and overall proportions. It is useful in crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the volume of the Deltoidal Icositetrahedron in cubic meters. The volume must be a positive number. The calculator will compute the corresponding short edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Icositetrahedron?
A: It is a Catalan solid with 24 congruent deltoid (kite-shaped) faces, 26 vertices, and 48 edges. It is the dual of the rhombicuboctahedron.

Q2: Are there different edge lengths in a Deltoidal Icositetrahedron?
A: Yes, the Deltoidal Icositetrahedron has two types of edges: short edges and long edges, with the short edges being the focus of this calculation.

Q3: What units should I use for volume?
A: The calculator expects volume in cubic meters, but you can use any consistent unit as long as you interpret the result in the same unit of length.

Q4: Can this formula be rearranged to find volume from the short edge?
A: Yes, the formula can be inverted to calculate volume if the short edge length is known, though the expression is more complex.

Q5: What are typical applications of this calculation?
A: This calculation is used in geometric modeling, 3D design, educational contexts, and in fields that study polyhedral structures.