Midsphere Radius Of Truncated Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \sqrt{\frac{TSA}{5 \times (\sqrt{3} + 6\sqrt{5 + 2\sqrt{5}})}} \]

Midsphere Radius (rₘ):

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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 the edges of the polyhedron. 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 mathematical formula:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \sqrt{\frac{TSA}{5 \times (\sqrt{3} + 6\sqrt{5 + 2\sqrt{5}})}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( TSA \) — Total Surface Area of the Truncated Dodecahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula derives from the geometric properties of the truncated dodecahedron and establishes the relationship between its total surface area and midsphere radius.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of polyhedrons, determining optimal packing arrangements, and analyzing geometric relationships in complex structures.

4. Using the Calculator

Tips: Enter the total surface area of the truncated dodecahedron in square meters. The value must be positive and 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 created 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 units should I use for input?
A: The calculator uses square meters for surface area input and returns meters for the midsphere radius result. Ensure consistent units throughout your calculations.

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

Q5: What is the typical range of values for midsphere radius?
A: The midsphere radius depends on the size of the polyhedron. For standard-sized truncated dodecahedrons, values typically range from a few centimeters to several meters, depending on the application.