Midsphere Radius of Truncated Dodecahedron Given Volume Calculator

Formula Used:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \left( \frac{12V}{5(99 + 47\sqrt{5})} \right)^{1/3} \]

Midsphere Radius of Truncated Dodecahedron (rₘ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Midsphere Radius of Truncated Dodecahedron?

The Midsphere Radius of a Truncated Dodecahedron is the radius of the sphere that is tangent to all edges of the truncated dodecahedron. It represents the sphere that fits perfectly within the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \left( \frac{12V}{5(99 + 47\sqrt{5})} \right)^{1/3} \]

Where:

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

Explanation: This formula derives the midsphere radius from the volume of a truncated dodecahedron using geometric relationships and mathematical constants specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of truncated dodecahedrons, determining optimal packing, and analyzing symmetry properties.

4. Using the Calculator

Tips: Enter the volume of the truncated dodecahedron 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 truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

Q2: How is midsphere radius different from insphere radius?
A: The midsphere is tangent to all edges, while the insphere is tangent to all faces. They represent different spheres within the polyhedron.

Q3: What are typical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and mathematical research involving polyhedral geometry.

Q4: Are there limitations to this formula?
A: This formula assumes a perfect truncated dodecahedron shape and may not apply to irregular or deformed polyhedrons.

Q5: Can this formula be derived from first principles?
A: Yes, the formula can be derived using geometric relationships, volume formulas, and the properties of regular polyhedrons in three-dimensional space.