Total Surface Area of Truncated Icosidodecahedron given Circumsphere Radius Calculator

Formula Used:

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

Total Surface Area (TSA):

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1. What is the Total Surface Area of Truncated Icosidodecahedron?

The Total Surface Area of a Truncated Icosidodecahedron is the total quantity of plane enclosed by the entire surface of this complex polyhedron. It's an important geometric property for understanding the spatial characteristics of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the total surface area based on the circumsphere radius, incorporating mathematical constants and square roots to account for the complex geometry of the truncated icosidodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material science, architectural design, and geometric analysis. It helps in determining material requirements, thermal properties, and structural characteristics of objects with this complex shape.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid. The calculator will compute the total surface area using the mathematical formula derived from geometric principles.

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: What is the Circumsphere Radius?
A: The circumsphere radius is the radius of the sphere that contains the polyhedron such that all vertices lie on the sphere's surface.

Q3: Why are there square roots in the formula?
A: The square roots account for the irrational numbers that naturally arise from the geometric relationships in regular polyhedra, particularly those involving the golden ratio.

Q4: What units should I use?
A: Use consistent units (typically meters for radius and square meters for area). The calculator assumes metric units.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula. The precision depends on the accuracy of the input value and the computational precision of the system.