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Total Surface Area Of Pentakis Dodecahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162\sqrt{5}} \times \left( \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \right)^2 \]

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1. What is Total Surface Area of Pentakis Dodecahedron?

The Total Surface Area of a Pentakis Dodecahedron is the total area of all its faces. A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron, featuring 60 isosceles triangular faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162\sqrt{5}} \times \left( \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \right)^2 \]

Where:

Explanation: The formula calculates the total surface area based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Pentakis Dodecahedron.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications in geometry, material science, and engineering, especially when dealing with the properties and behavior of polyhedral structures.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero to compute the total surface area accurately.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid with 60 faces, each being an isosceles triangle. It is the dual of the truncated icosahedron.

Q2: How is the insphere radius defined?
A: The insphere radius is the radius of the largest sphere that can fit inside the Pentakis Dodecahedron, touching all its faces.

Q3: What units should be used for input?
A: The input for insphere radius should be in meters (m), and the output total surface area will be in square meters (m²).

Q4: Are there limitations to this formula?
A: This formula is specific to the Pentakis Dodecahedron and assumes ideal geometric conditions. It may not apply to modified or irregular shapes.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Pentakis Dodecahedron. Other polyhedra require different formulas for surface area calculation.

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