Total Surface Area of Truncated Cuboctahedron given Circumsphere Radius Calculator

Formula Used:

\[ TSA = 12 \times (2 + \sqrt{2} + \sqrt{3}) \times \left( \frac{2 \times r_c}{\sqrt{13 + 6\sqrt{2}}} \right)^2 \]

Total Surface Area of Truncated Cuboctahedron:

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

The Total Surface Area of a Truncated Cuboctahedron is the total quantity of plane enclosed by the entire surface of this Archimedean solid. It's calculated based on the circumsphere radius using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 12 \times (2 + \sqrt{2} + \sqrt{3}) \times \left( \frac{2 \times r_c}{\sqrt{13 + 6\sqrt{2}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Truncated Cuboctahedron (m²)
\( r_c \) — Circumsphere Radius of Truncated Cuboctahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{3} \) — Square root of 3 (approximately 1.7321)

Explanation: This formula derives from the geometric properties of the truncated cuboctahedron and its relationship with the circumscribed sphere.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is crucial for various applications including material science, architectural design, and geometric analysis of this complex polyhedron.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Cuboctahedron?
A: A truncated cuboctahedron is an Archimedean solid with 26 faces: 12 squares, 8 regular hexagons, and 6 regular octagons.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the truncated cuboctahedron such that all vertices lie on the sphere's surface.

Q3: Why are square roots used in the formula?
A: The square roots come from the geometric relationships and trigonometric properties inherent in the polyhedron's structure.

Q4: What are typical applications of this calculation?
A: This calculation is used in crystallography, architectural design, 3D modeling, and mathematical research involving polyhedra.

Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect truncated cuboctahedron and provides precise results when correct input values are used.