Total Surface Area of Truncated Icosidodecahedron given Midsphere Radius Calculator

Formula Used:

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

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 represents the sum of the areas of all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

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

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is important for various applications including material science, architectural design, and mathematical modeling of complex polyhedral structures.

4. Using the Calculator

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

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 Midsphere Radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: What units should I use for input?
A: The calculator expects the midsphere radius in meters, and returns the surface area in square meters.

Q4: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: Is this formula applicable to all polyhedra?
A: No, this specific formula is only valid for calculating the total surface area of a truncated icosidodecahedron given its midsphere radius.