Insphere Radius of Icosahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \left( \frac{12 \cdot V}{5 \cdot (3 + \sqrt{5})} \right)^{1/3} \]

Insphere Radius of Icosahedron (rᵢ):

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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 within the Icosahedron such that it touches all the faces of the polyhedron. It represents the largest sphere that can fit inside the Icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \left( \frac{12 \cdot V}{5 \cdot (3 + \sqrt{5})} \right)^{1/3} \]

Where:

\( r_i \) — Insphere Radius of Icosahedron (m)
\( V \) — Volume of Icosahedron (m³)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula derives the insphere radius from the volume of a regular icosahedron, using mathematical constants and cube root operations.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry, material science, and engineering applications where understanding the internal spatial relationships of polyhedral structures is required.

4. Using the Calculator

Tips: Enter the volume of the icosahedron in cubic meters. The volume must be a positive value greater than zero for accurate calculation.

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 this formula derived?
A: The formula is derived from the geometric relationships between the volume, insphere radius, and other dimensional properties of a regular icosahedron.

Q3: What are the units for the insphere radius?
A: The insphere radius is measured in meters (m), consistent with the volume input in cubic meters (m³).

Q4: Can this calculator handle very large or small values?
A: Yes, the calculator can process a wide range of volume values, though extremely large or small values may be limited by PHP's floating-point precision.

Q5: Are there other ways to calculate the insphere radius?
A: Yes, the insphere radius can also be calculated from the edge length or other dimensional properties of the icosahedron using different formulas.