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Volume of Triakis Icosahedron Calculator

Volume of Triakis Icosahedron Formula:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times (l_e)^3 \]

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1. What is Volume of Triakis Icosahedron?

The Volume of Triakis Icosahedron is the quantity of three dimensional space enclosed by the entire surface of Triakis Icosahedron. It is a Catalan solid that can be derived from an icosahedron by adding pyramids to each face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times (l_e)^3 \]

Where:

Explanation: The formula calculates the volume based on the edge length of the underlying icosahedron, incorporating the mathematical constant √5 which is characteristic of icosahedral geometry.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields. For the Triakis Icosahedron, volume calculation helps in material estimation, structural analysis, and understanding spatial properties of this complex polyhedron.

4. Using the Calculator

Tips: Enter the icosahedral edge length in meters. The value must be positive and non-zero. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that consists of 60 isosceles triangles. It can be constructed by adding a triangular pyramid to each face of a regular icosahedron.

Q2: How accurate is this calculation?
A: The calculation is mathematically precise, using the exact formula derived from geometric principles. The result is accurate to the precision of the input values.

Q3: Can I use different units?
A: Yes, but ensure consistency. If you input edge length in centimeters, the volume will be in cubic centimeters. The calculator displays results in cubic meters by default.

Q4: What are the applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and anywhere where the geometric properties of a Triakis Icosahedron need to be analyzed.

Q5: Why does the formula contain √5?
A: The square root of 5 appears naturally in the geometry of regular icosahedra and related polyhedra due to their relationship with the golden ratio.

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