Space Diagonal of Icosahedron given Insphere Radius Calculator

Formula Used:

\[ d_{Space} = \sqrt{10 + (2 \times \sqrt{5})} \times \frac{6 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \]

Space Diagonal of Icosahedron:

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1. What is Space Diagonal of Icosahedron?

The Space Diagonal of Icosahedron is the line connecting two vertices that are not on the same face of Icosahedron. It represents the longest distance between any two vertices in this regular polyhedron with 20 faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_{Space} = \sqrt{10 + (2 \times \sqrt{5})} \times \frac{6 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \]

Where:

\( d_{Space} \) — Space Diagonal of Icosahedron (m)
\( r_i \) — Insphere Radius of Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula derives from the geometric properties of a regular icosahedron, relating the space diagonal to the radius of the inscribed sphere.

3. Importance of Space Diagonal Calculation

Details: Calculating the space diagonal is important in geometry, architecture, and 3D modeling for determining the maximum dimensions and spatial relationships within icosahedral structures.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding space diagonal of the icosahedron.

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: How is insphere radius defined for an icosahedron?
A: The insphere radius is the radius of the largest sphere that can be contained within the icosahedron, tangent to all its faces.

Q3: What are typical applications of icosahedron geometry?
A: Icosahedrons are used in molecular modeling (fullerenes), geodesic domes, game design, and various architectural structures.

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

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for regular icosahedrons, with accuracy limited only by the precision of the input values and computational rounding.