Total Surface Area of Truncated Dodecahedron given Circumsphere Radius Calculator

Formula Used:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{4 \times r_c}{\sqrt{74 + 30 \times \sqrt{5}}} \right)^2 \]

Total Surface Area of Truncated Dodecahedron:

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1. What is the 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 the Truncated Dodecahedron. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 5 \times (\sqrt{3} + 6 \times \sqrt{5 + 2 \times \sqrt{5}}) \times \left( \frac{4 \times r_c}{\sqrt{74 + 30 \times \sqrt{5}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Truncated Dodecahedron (m²)
\( r_c \) — Circumsphere Radius of Truncated Dodecahedron (m)

Explanation: This formula calculates the total surface area based on the circumsphere radius of the truncated dodecahedron, incorporating various mathematical constants and operations.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is crucial for understanding the geometric properties of truncated dodecahedrons, which have applications in architecture, material science, and mathematical modeling.

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

Q2: What is the Circumsphere Radius?
A: The circumsphere radius is the radius of the sphere that contains the truncated dodecahedron in such a way that all the vertices are lying on the sphere.

Q3: Why is this calculation important?
A: Surface area calculations are fundamental in various fields including material science, architecture, and engineering where geometric properties need to be quantified.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for perfect truncated dodecahedrons and assumes ideal geometric conditions.

Q5: What units should be used?
A: The calculator uses meters for input and square meters for output, but the formula works with any consistent unit system.