Volume Of Triakis Icosahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left( \frac{11 \times TSA}{15 \times \sqrt{109 - 30\sqrt{5}}} \right)^{\frac{3}{2}} \]

Volume of Triakis Icosahedron:

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1. What is the Volume of Triakis Icosahedron?

The volume of a Triakis Icosahedron is the amount of three-dimensional space enclosed by its surface. It is a Catalan solid that can be derived from the icosahedron by adding pyramids to each face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{5}{44} \times (5 + 7\sqrt{5}) \times \left( \frac{11 \times TSA}{15 \times \sqrt{109 - 30\sqrt{5}}} \right)^{\frac{3}{2}} \]

Where:

\( V \) — Volume of Triakis Icosahedron (m³)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume of a Triakis Icosahedron based on its total surface area, 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 various fields including mathematics, engineering, architecture, and 3D modeling. It helps in understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive. The calculator will compute the corresponding 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 isosceles triangular faces.

Q2: What are the applications of this calculation?
A: This calculation is useful in geometry education, 3D modeling, architectural design, and any application requiring precise volume measurements of complex polyhedra.

Q3: How accurate is this formula?
A: The formula is mathematically exact for perfect Triakis Icosahedrons. The accuracy of the result depends on the precision of the input surface area value.

Q4: Can this calculator handle different units?
A: The calculator expects input in square meters and outputs volume in cubic meters. For other units, convert your measurements to square meters before calculation.

Q5: What is the range of valid input values?
A: The total surface area must be a positive number. There is no theoretical upper limit, but extremely large values may exceed computational precision.