Midsphere Radius of Icosahedron given Circumsphere Radius Calculator

Midsphere Radius of Icosahedron Formula:

\[ r_m = \frac{(1 + \sqrt{5}) \times r_c}{\sqrt{10 + (2 \times \sqrt{5})}} \]

Midsphere Radius of Icosahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Midsphere Radius of Icosahedron?

The Midsphere Radius of Icosahedron is defined as the radius of the sphere for which all the edges of the Icosahedron become a tangent line on that sphere. It's an important geometric property of this regular polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_m = \frac{(1 + \sqrt{5}) \times r_c}{\sqrt{10 + (2 \times \sqrt{5})}} \]

Where:

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

Explanation: This formula establishes the precise mathematical relationship between the circumsphere radius and the midsphere radius of a regular icosahedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is crucial for geometric analysis, 3D modeling, and understanding the spatial properties of icosahedrons in various applications including architecture, chemistry, and computer graphics.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding midsphere radius using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

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

Q2: What's the difference between circumsphere and midsphere?
A: The circumsphere passes through all vertices, while the midsphere is tangent to all edges of the icosahedron.

Q3: Can this formula be used for irregular icosahedrons?
A: No, this specific formula applies only to regular icosahedrons where all faces are equilateral triangles.

Q4: What are practical applications of this calculation?
A: Used in molecular modeling (fullerenes), geodesic dome design, and computer graphics for generating spherical approximations.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular icosahedrons, with precision limited only by computational floating-point arithmetic.