Total Surface Area Of Triakis Icosahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{4 \times r_i}{\sqrt{\frac{10 \times (33 + 13\sqrt{5})}{61}}} \right)^2 \]

Total Surface Area of Triakis Icosahedron:

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1. What is the Total Surface Area of Triakis Icosahedron?

The Total Surface Area of a Triakis Icosahedron refers to the total area of all its faces combined. A Triakis Icosahedron is a polyhedron created by attaching a triangular pyramid to each face of a regular icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{11} \times \sqrt{109 - 30\sqrt{5}} \times \left( \frac{4 \times r_i}{\sqrt{\frac{10 \times (33 + 13\sqrt{5})}{61}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Triakis Icosahedron (m²)
\( r_i \) — Insphere Radius of Triakis Icosahedron (m)

Explanation: This formula calculates the total surface area based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Triakis Icosahedron.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is essential in geometry and various practical applications, including material estimation, structural design, and understanding the properties of polyhedra.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid formed by adding a triangular pyramid to each face of a regular icosahedron, resulting in a polyhedron with 60 faces.

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

Q3: What units should be used for input?
A: The calculator uses meters for length units, resulting in square meters for area. Ensure consistent units for accurate results.

Q4: Are there limitations to this formula?
A: This formula is specific to the Triakis Icosahedron geometry and assumes ideal mathematical conditions.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Triakis Icosahedron. Other polyhedra require different formulas.