Volume Of Triakis Tetrahedron Given Insphere Radius Calculator

Volume of Triakis Tetrahedron Formula:

\[ V = \frac{3}{20} \times \sqrt{2} \times \left( \frac{4 \times r_i \times \sqrt{11}}{3 \times \sqrt{2}} \right)^3 \]

Volume of Triakis Tetrahedron:

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1. What is the Volume of Triakis Tetrahedron?

The volume of a Triakis Tetrahedron is the measure of three-dimensional space enclosed by its surface. It is a Catalan solid that can be formed by attaching a triangular pyramid to each face of a regular tetrahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{3}{20} \times \sqrt{2} \times \left( \frac{4 \times r_i \times \sqrt{11}}{3 \times \sqrt{2}} \right)^3 \]

Where:

\( V \) — Volume of Triakis Tetrahedron (m³)
\( r_i \) — Insphere Radius of Triakis Tetrahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{11} \) — Square root of 11 (approximately 3.3166)

Explanation: This formula calculates the volume of a Triakis Tetrahedron based on its insphere radius, using mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and architecture. For Triakis Tetrahedrons, volume calculation helps in material estimation, structural analysis, and understanding spatial properties of this specific polyhedral form.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero. The calculator will compute the volume using the mathematical formula shown above.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Tetrahedron?
A: A Triakis Tetrahedron is a Catalan solid that results from attaching a triangular pyramid to each face of a regular tetrahedron. It has 12 faces, 18 edges, and 8 vertices.

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

Q3: Can this formula be used for any Triakis Tetrahedron?
A: Yes, this formula applies to all regular Triakis Tetrahedrons where the base tetrahedron and the attached pyramids are regular.

Q4: What are the units of measurement?
A: The input should be in meters and the output volume will be in cubic meters (m³). Consistent units must be used throughout.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for the given formula. The accuracy of the result depends on the precision of the input value and the computational precision of the calculator.