Circumsphere Radius of Truncated Icosidodecahedron given Total Surface Area Calculator

Formula Used:

\[ r_c = \frac{\sqrt{31 + 12\sqrt{5}}}{2} \times \sqrt{\frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Truncated Icosidodecahedron?

The Circumsphere Radius of a Truncated Icosidodecahedron is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface. It is a key geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{31 + 12\sqrt{5}}}{2} \times \sqrt{\frac{TSA}{30 \times (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}})}} \]

Where:

\( r_c \) — Circumsphere radius (m)
\( TSA \) — Total surface area (m²)
\( \sqrt{5} \) — Square root of 5 (≈2.23607)
\( \sqrt{3} \) — Square root of 3 (≈1.73205)

Explanation: This formula derives the circumsphere radius from the total surface area using the geometric properties of the truncated icosidodecahedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions and bounding sphere of the polyhedron, which has applications in crystallography, molecular modeling, and geometric analysis.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and non-zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Icosidodecahedron?
A: A truncated icosidodecahedron is an Archimedean solid with 120 vertices, 180 edges, and 62 faces (30 squares, 20 hexagons, and 12 decagons).

Q2: Why is this formula complex?
A: The formula incorporates multiple square roots and constants that arise from the complex geometry and symmetry of the truncated icosidodecahedron.

Q3: What are typical values for circumsphere radius?
A: The circumsphere radius depends on the size of the polyhedron. For a polyhedron with TSA around 100m², the circumsphere radius would be approximately 2-3 meters.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the truncated icosidodecahedron. Other polyhedra have different formulas for circumsphere radius calculation.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 10 decimal places precision, which is sufficient for most geometric and engineering applications.