Circumsphere Radius of Icosahedron given Insphere Radius Calculator

Formula Used:

\[ r_c = \frac{\sqrt{10 + (2 \times \sqrt{5})}}{4} \times \frac{12 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \]

Circumsphere Radius of Icosahedron (r_c):

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1. What is the Circumsphere Radius of Icosahedron?

The Circumsphere Radius of an Icosahedron is the radius of the sphere that contains the Icosahedron in such a way that all the vertices are lying on the sphere. It is a key geometric property of this regular polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{10 + (2 \times \sqrt{5})}}{4} \times \frac{12 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \]

Where:

\( r_c \) — Circumsphere Radius of Icosahedron (m)
\( r_i \) — Insphere Radius of Icosahedron (m)

Explanation: This formula establishes the mathematical relationship between the circumsphere radius and the insphere radius of a regular icosahedron, using fundamental geometric constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential in geometry, 3D modeling, and various engineering applications where the spatial dimensions and proportions of icosahedral structures need to be determined.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumsphere and insphere radii?
A: The circumsphere radius is always larger than the insphere radius in a regular icosahedron, with a fixed mathematical relationship between them.

Q2: Can this formula be used for irregular icosahedrons?
A: No, this formula is specifically derived for regular icosahedrons where all faces are equilateral triangles and all vertices lie on both spheres.

Q3: What are typical values for these radii?
A: The values depend on the size of the icosahedron. For a regular icosahedron with edge length a, the circumsphere radius is approximately 0.951a and the insphere radius is approximately 0.755a.

Q4: Why are there square roots in the formula?
A: The square roots come from the geometric properties of the regular icosahedron and the mathematical relationships between its various dimensions.

Q5: What practical applications use this calculation?
A: This calculation is used in molecular modeling, architectural design, geodesic dome construction, and various fields where icosahedral symmetry is important.