Insphere Radius of Icosahedron given Circumsphere Radius Calculator

Formula Used:

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

Insphere Radius of Icosahedron (r_i):

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1. What is the Insphere Radius of Icosahedron?

The Insphere Radius of an Icosahedron is the radius of the sphere that is contained by the Icosahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the Icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_i \) — Insphere Radius of Icosahedron (m)
\( r_c \) — Circumsphere Radius of Icosahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the insphere radius based on the given circumsphere radius of a regular icosahedron, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling for understanding the spatial relationships within polyhedra, determining packing efficiency, and solving various geometric problems involving regular icosahedrons.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius using the mathematical formula.

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: How is the circumsphere radius related to the insphere radius?
A: The circumsphere radius is the distance from the center to any vertex, while the insphere radius is the distance from the center to the center of any face. They have a fixed mathematical relationship in a regular icosahedron.

Q3: What are typical applications of this calculation?
A: This calculation is used in geometry, crystallography, molecular modeling, and various engineering applications where regular icosahedral structures are encountered.

Q4: Are there limitations to this formula?
A: This formula applies only to regular icosahedrons. For irregular or modified icosahedral structures, different calculations would be required.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit, but you can convert your measurements to meters before inputting them, and the result will be in meters as well.