Midsphere Radius of Truncated Icosidodecahedron given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}}}{2} \times \sqrt{\frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}} \]

Midsphere Radius (rₘ):

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1. What is Midsphere Radius of Truncated Icosidodecahedron?

The Midsphere Radius of a Truncated Icosidodecahedron is the radius of the sphere that is tangent to all edges of the polyhedron. It represents the sphere that touches every edge of the Truncated Icosidodecahedron exactly at one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{\sqrt{30 + 12\sqrt{5}}}{2} \times \sqrt{\frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}} \]

Where:

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

Explanation: This formula derives the midsphere radius from the total surface area using mathematical constants and geometric relationships specific to the Truncated Icosidodecahedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties and proportions of the Truncated Icosidodecahedron. It helps in various applications including architectural design, molecular modeling, and mathematical research.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding midsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Icosidodecahedron?
A: A Truncated Icosidodecahedron is an Archimedean solid with 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons.

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

Q3: What units should I use for input?
A: Use square meters for surface area. The result will be in meters for the radius.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Truncated Icosidodecahedron due to its unique geometric properties.

Q5: What if I get an error in calculation?
A: Ensure the input value is positive and valid. The formula requires positive surface area values greater than zero.