Edge Length Of Icosahedron Given Insphere Radius Calculator

Formula Used:

\[ Edge\ Length\ of\ Icosahedron = \frac{12 \times Insphere\ Radius\ of\ Icosahedron}{\sqrt{3} \times (3 + \sqrt{5})} \]

Edge Length of Icosahedron:

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

The edge length of an icosahedron can be calculated from its insphere radius using a specific mathematical formula. This relationship is important in geometry for determining the size of a regular icosahedron when the radius of its inscribed sphere is known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Edge\ Length = \frac{12 \times Insphere\ Radius}{\sqrt{3} \times (3 + \sqrt{5})} \]

Where:

Insphere Radius — Radius of the sphere inscribed in the icosahedron (m)
Edge Length — Length of any edge of the icosahedron (m)

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

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the insphere radius is crucial in geometry, 3D modeling, and various engineering applications where precise dimensions of icosahedral structures are required.

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 edge length of the icosahedron.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is this specific formula used?
A: This formula is derived from the geometric properties of a regular icosahedron and provides the exact relationship between its insphere radius and edge length.

Q3: Can this calculator be used for irregular icosahedrons?
A: No, this calculator is specifically designed for regular icosahedrons where all edges are equal and all faces are congruent equilateral triangles.

Q4: What are practical applications of this calculation?
A: This calculation is used in architecture, molecular modeling, geodesic dome design, and various fields where icosahedral structures are employed.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular icosahedrons. The accuracy depends on the precision of the input insphere radius value.