Short Edge of Hexakis Icosahedron Given Midsphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{8 \times r_m}{5 + 3\sqrt{5}} \]

Short Edge of Hexakis Icosahedron:

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1. What is the Short Edge of Hexakis Icosahedron?

The Short Edge of Hexakis Icosahedron is the length of the shortest edge that connects two adjacent vertices of the Hexakis Icosahedron. It is a fundamental geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \frac{8 \times r_m}{5 + 3\sqrt{5}} \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the shortest edge length based on the given midsphere radius, incorporating mathematical constants and geometric relationships specific to the Hexakis Icosahedron.

3. Importance of Short Edge Calculation

Details: Accurate calculation of the short edge is crucial for geometric modeling, structural analysis, and various applications in mathematics, architecture, and engineering where precise dimensions of polyhedra are required.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding short edge length of the Hexakis Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is defined as the radius of the sphere for which all the edges of the Hexakis Icosahedron become a tangent line on that sphere.

Q3: Are there other ways to calculate the short edge?
A: Yes, the short edge can also be calculated using other geometric properties such as volume or surface area, but the midsphere radius provides a direct relationship.

Q4: What are typical values for the short edge?
A: The short edge length varies depending on the size of the Hexakis Icosahedron, but it is always the smallest edge length in the polyhedron.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific formula and calculator are designed exclusively for the Hexakis Icosahedron geometry.