Insphere Radius of Hexakis Icosahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \left( \frac{88V}{25\sqrt{6(185 + 82\sqrt{5})}} \right)^{\frac{1}{3}} \]

Insphere Radius of Hexakis Icosahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Insphere Radius of Hexakis Icosahedron?

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

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ r_i = \frac{\sqrt{\frac{15}{241} \cdot (275 + 119\sqrt{5})}}{4} \cdot \left( \frac{88V}{25\sqrt{6(185 + 82\sqrt{5})}} \right)^{\frac{1}{3}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( V \) — Volume of Hexakis Icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the insphere radius based on the volume of the Hexakis Icosahedron, using geometric relationships and mathematical constants.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and engineering applications where understanding the internal dimensions and properties of complex polyhedra is required for design and analysis purposes.

4. Using the Calculator

Tips: Enter the volume of the Hexakis Icosahedron in cubic meters. The value must be positive and valid for accurate calculation of the insphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: What units should I use for volume input?
A: The calculator expects volume input in cubic meters (m³). Make sure to convert from other units if necessary.

Q3: Can this calculator handle very large or very small volumes?
A: Yes, the calculator can handle a wide range of volume values, but extremely large or small values may affect computational precision.

Q4: What is the significance of the insphere radius?
A: The insphere radius helps determine the maximum size of a sphere that can be inscribed within the polyhedron, which is useful in various geometric and engineering applications.

Q5: Are there any limitations to this calculation?
A: The calculation assumes a perfect Hexakis Icosahedron shape and may not account for manufacturing tolerances or imperfections in real-world objects.