Total Surface Area Of Triakis Tetrahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{3}{5} \times \sqrt{11} \times \frac{11}{2} \times \frac{16}{9} \times r_i^2 \]

Total Surface Area (TSA):

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

The Total Surface Area of a Triakis Tetrahedron is the total area of all its faces. A Triakis Tetrahedron is a Catalan solid that can be obtained by adding a triangular pyramid to each face of a regular tetrahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{3}{5} \times \sqrt{11} \times \frac{11}{2} \times \frac{16}{9} \times r_i^2 \]

Where:

\( TSA \) — Total Surface Area of Triakis Tetrahedron (m²)
\( r_i \) — Insphere Radius of Triakis Tetrahedron (m)
\( \sqrt{11} \) — Square root of 11

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

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is fundamental in various fields including architecture, engineering, and materials science. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the total surface area using the provided formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Tetrahedron?
A: A Triakis Tetrahedron is a Catalan solid with 12 isosceles triangular faces. It is the dual polyhedron of the truncated tetrahedron.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, tangent to all its faces.

Q3: Are there other ways to calculate surface area?
A: Yes, surface area can also be calculated using edge length or other geometric parameters, but this calculator specifically uses the insphere radius.

Q4: What are typical applications of this calculation?
A: This calculation is useful in crystallography, molecular modeling, and the design of geometric structures in architecture and engineering.

Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect Triakis Tetrahedron. Real-world applications may require adjustments for manufacturing tolerances and material properties.