1. What is Volume Of Hexakis Icosahedron Given Insphere Radius?

The volume of a Hexakis Icosahedron given its insphere radius is the total three-dimensional space enclosed by the surface of this complex polyhedron, calculated based on the radius of the sphere that touches all its faces internally.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

Where:

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

Explanation: This formula relates the volume of the polyhedron to the radius of its inscribed sphere through a complex mathematical relationship involving square roots and cube powers.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in mathematics, engineering, architecture, and various scientific fields where spatial measurements and three-dimensional analysis 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 volume of the Hexakis Icosahedron.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron and touch all its faces.

Q3: What are the units of measurement?
A: The calculator uses meters for input (insphere radius) and cubic meters for output (volume).

Q4: How accurate is the calculation?
A: The calculation provides results with 6 decimal places precision, suitable for most mathematical and engineering applications.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Hexakis Icosahedron and its relationship between volume and insphere radius.