Insphere Radius of Rhombic Triacontahedron Given Volume Calculator

Formula Used:

\[ r_i = \left( \frac{V}{4 \cdot \sqrt{5 + 2 \cdot \sqrt{5}}} \right)^{\frac{1}{3}} \cdot \sqrt{\frac{5 + 2 \cdot \sqrt{5}}{5}} \]

Insphere Radius of Rhombic Triacontahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Insphere Radius of Rhombic Triacontahedron?

The Insphere Radius of Rhombic Triacontahedron is the radius of the sphere that is contained by the Rhombic Triacontahedron in such a way that all the faces are just touching the sphere. It represents the largest sphere that can fit inside the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \left( \frac{V}{4 \cdot \sqrt{5 + 2 \cdot \sqrt{5}}} \right)^{\frac{1}{3}} \cdot \sqrt{\frac{5 + 2 \cdot \sqrt{5}}{5}} \]

Where:

\( r_i \) — Insphere Radius (m)
\( V \) — Volume of Rhombic Triacontahedron (m³)
\( \sqrt{} \) — Square root function

Explanation: The formula derives the insphere radius from the volume of the Rhombic Triacontahedron using geometric relationships and mathematical constants related to this specific polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and materials science for understanding the spatial properties of the Rhombic Triacontahedron, including its packing efficiency and internal volume characteristics.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombic Triacontahedron?
A: A Rhombic Triacontahedron is a convex polyhedron with 30 rhombic faces. It is one of the Catalan solids and is the dual polyhedron of the icosidodecahedron.

Q2: What units should I use for volume input?
A: The calculator expects volume in cubic meters (m³). If your volume is in different units, convert it to cubic meters before calculation.

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 be limited by PHP's floating-point precision.

Q4: What is the geometric significance of the insphere radius?
A: The insphere radius indicates the maximum size of a sphere that can be inscribed within the polyhedron, touching all faces internally.

Q5: Are there any limitations to this calculation?
A: The calculation assumes a perfect Rhombic Triacontahedron shape and may not account for manufacturing tolerances or deformations in physical objects.