1. What is Midsphere Radius of Icosahedron?

The midsphere radius of an icosahedron is the radius of the sphere that is tangent to all the edges of the icosahedron. For a regular icosahedron, this sphere touches each edge at its midpoint.

2. How Does the Calculator Work?

The calculator uses the midsphere radius formula:

Where:

• \( r_m \) — Midsphere radius
• \( P \) — Perimeter of the icosahedron
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the radius of the sphere that touches all edges at their midpoints based on the given perimeter of the icosahedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of icosahedrons and their relationship with inscribed spheres.

4. Using the Calculator

Tips: Enter the perimeter of the icosahedron in the input field. The value must be a positive number. The calculator will compute the midsphere radius based on the mathematical formula.

5. Frequently Asked Questions (FAQ)

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

Q2: How is midsphere radius different from insphere radius?
A: The insphere radius touches the faces of the polyhedron, while the midsphere radius touches the edges at their midpoints.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometry, 3D modeling, molecular structures, and architectural design involving icosahedral shapes.

Q4: Can this formula be used for irregular icosahedrons?
A: No, this formula is specifically for regular icosahedrons where all faces are equilateral triangles and all vertices are equivalent.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact for regular icosahedrons, though the displayed result may be rounded for practical purposes.