Volume Of Hexakis Icosahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \frac{25}{88} \times \sqrt{6(185+82\sqrt{5})} \times \left( \frac{44 \times TSA}{15 \times \sqrt{10(417+107\sqrt{5})}} \right)^{\frac{3}{2}} \]

Volume of Hexakis Icosahedron:

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1. What is the Volume of Hexakis Icosahedron?

The Volume of Hexakis Icosahedron is the quantity of three dimensional space enclosed by the entire surface of Hexakis Icosahedron. It is calculated based on the total surface area using a specific mathematical formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{25}{88} \times \sqrt{6(185+82\sqrt{5})} \times \left( \frac{44 \times TSA}{15 \times \sqrt{10(417+107\sqrt{5})}} \right)^{\frac{3}{2}} \]

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( TSA \) — Total Surface Area of Hexakis Icosahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume of a Hexakis Icosahedron based on its total surface area, using mathematical constants derived from the geometry of the shape.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in various fields including mathematics, engineering, architecture, and 3D modeling. It helps in understanding the spatial properties and capacity of the shape.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding volume of the Hexakis Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: Why is this formula so complex?
A: The formula involves mathematical constants derived from the specific geometry and proportions of the Hexakis Icosahedron, which includes irrational numbers like √5.

Q3: What units should I use for input?
A: The calculator expects the total surface area in square meters, and returns the volume in cubic meters. Make sure to use consistent units.

Q4: Can this calculator handle very large or very small values?
A: The calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating point precision.

Q5: Is this formula accurate for all Hexakis Icosahedrons?
A: Yes, this formula is mathematically derived and provides accurate volume calculations for any regular Hexakis Icosahedron given its total surface area.