Volume Of Triakis Icosahedron Given Midsphere Radius Calculator

Formula Used:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left(\frac{4r_m}{1 + \sqrt{5}}\right)^3 \]

Volume of Triakis Icosahedron:

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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's a Catalan solid that can be obtained by adding pyramids to each face of a regular icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left(\frac{4r_m}{1 + \sqrt{5}}\right)^3 \]

Where:

\( V \) — Volume of Triakis Icosahedron (m³)
\( r_m \) — Midsphere Radius of Triakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the volume based on the midsphere radius, incorporating the mathematical constant √5 which is fundamental in icosahedral geometry.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is crucial in various fields including architecture, engineering, material science, and 3D modeling. It helps in determining material requirements, structural properties, and spatial relationships.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Triakis Icosahedron?
A: A Triakis Icosahedron is a Catalan solid that results from adding a triangular pyramid to each face of a regular icosahedron, creating a polyhedron with 60 faces.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: Can this calculator handle different units?
A: The calculator assumes meters as input. For other units, convert your measurement to meters first, then convert the result back to your desired unit.

Q4: What is the precision of the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most practical applications.

Q5: Are there any limitations to this formula?
A: This formula is specifically designed for a perfect Triakis Icosahedron. It assumes ideal geometric conditions and may not apply to irregular or deformed shapes.