1. What is the Edge Length of Icosahedron given Midsphere Radius?

The edge length of an icosahedron can be calculated from its midsphere radius using the mathematical relationship between these two geometric properties of the regular icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( l_e \) — Edge Length of Icosahedron (m)
• \( r_m \) — Midsphere Radius of Icosahedron (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the precise mathematical relationship between the midsphere radius and the edge length in a regular icosahedron, one of the five Platonic solids.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the midsphere radius is essential in geometry, 3D modeling, crystallography, and various engineering applications where icosahedral structures are used.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid for accurate calculation of the edge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular icosahedron?
A: A regular icosahedron is a convex polyhedron with 20 identical equilateral triangle faces, 30 edges, and 12 vertices.

Q2: What is the midsphere radius of an icosahedron?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the icosahedron.

Q3: Are there other ways to calculate edge length?
A: Yes, the edge length can also be calculated from other parameters such as circumsphere radius, insphere radius, surface area, or volume.

Q4: What are practical applications of this calculation?
A: This calculation is used in molecular modeling, architectural design, game development, and materials science where icosahedral structures appear.

Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect regular icosahedron and provides precise results when correct input values are used.