Total Surface Area of Truncated Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{4 \times r_m}{5 + 3 \times \sqrt{5}} \right)^2 \]

Total Surface Area of Truncated Dodecahedron:

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1. What is Total Surface Area of Truncated Dodecahedron?

The Total Surface Area of a Truncated Dodecahedron is the total quantity of plane enclosed by the entire surface of this polyhedron. It's an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{4 \times r_m}{5 + 3 \times \sqrt{5}} \right)^2 \]

Where:

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

Explanation: This formula calculates the total surface area based on the midsphere radius, incorporating mathematical constants and geometric relationships specific to the truncated dodecahedron.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, structural analysis, and geometric modeling of truncated dodecahedron shapes in architecture and engineering.

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 dodecahedron?
A: A truncated dodecahedron is an Archimedean solid obtained by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

Q2: What is midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the polyhedron.

Q3: What units should I use for input?
A: The calculator uses meters for input, but you can use any consistent unit system as the formula is dimensionally consistent.

Q4: Can this calculator handle very large or small values?
A: Yes, the calculator can handle a wide range of positive values, though extremely large values may be limited by computational precision.

Q5: Is this formula accurate for all truncated dodecahedrons?
A: Yes, this formula is mathematically exact and applies to all regular truncated dodecahedrons.