Midsphere Radius of Truncated Icosahedron given Volume Calculator

Midsphere Radius of Truncated Icosahedron Formula:

\[ r_m = \frac{3(1+\sqrt{5})}{4} \times \sqrt[3]{\frac{4V}{125+43\sqrt{5}}} \]

Midsphere Radius of Truncated Icosahedron (rₘ):

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1. What is the Midsphere Radius of Truncated Icosahedron?

The Midsphere Radius of a Truncated Icosahedron is the radius of the sphere that is tangent to all the edges of the polyhedron. For a truncated icosahedron (soccer ball shape), this sphere touches each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{3(1+\sqrt{5})}{4} \times \sqrt[3]{\frac{4V}{125+43\sqrt{5}}} \]

Where:

\( r_m \) — Midsphere radius (m)
\( V \) — Volume of truncated icosahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula derives from the geometric properties of the truncated icosahedron, relating the midsphere radius to the volume through mathematical constants specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry, materials science, and architecture for understanding the spatial properties and packing efficiency of truncated icosahedral structures.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated icosahedron?
A: A truncated icosahedron is an Archimedean solid with 12 regular pentagonal faces, 20 regular hexagonal faces, 90 edges, and 60 vertices. It's the shape of a soccer ball.

Q2: How is the midsphere different from the insphere?
A: The midsphere is tangent to all edges, while the insphere is tangent to all faces. For a truncated icosahedron, these spheres have different radii.

Q3: What are practical applications of this calculation?
A: This calculation is used in nanotechnology (fullerenes), architectural design, and materials science where truncated icosahedral structures occur.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated icosahedra. Other polyhedra have different formulas for their midsphere radii.

Q5: What units should be used for volume input?
A: The calculator expects volume in cubic meters, but you can use any consistent unit system as long as the output radius is interpreted in the same length units.