Total Surface Area Of Hexakis Icosahedron Given Volume Calculator

Formula Used:

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

Total Surface Area (TSA):

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1. What is the Total Surface Area of Hexakis Icosahedron?

The Total Surface Area of a Hexakis Icosahedron is the total area of all its faces. A Hexakis Icosahedron is a polyhedron with 120 triangular faces, 90 edges, and 32 vertices. It is the dual of the truncated dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula calculates the total surface area based on the volume of the Hexakis Icosahedron, incorporating mathematical constants and operations.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is important in various fields such as geometry, architecture, and material science. It helps in understanding the properties and applications of the polyhedron.

4. Using the Calculator

Tips: Enter the volume of the Hexakis Icosahedron in cubic meters. The volume must be a positive number.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid with 120 faces, 90 edges, and 32 vertices. It is the dual of the truncated dodecahedron.

Q2: Why is the surface area important?
A: The surface area is crucial for applications involving coating, painting, or material usage where the amount of surface coverage is needed.

Q3: Can the calculator handle very large or very small volumes?
A: Yes, as long as the volume is a positive number, the calculator can compute the corresponding surface area.

Q4: Are there any limitations to this formula?
A: The formula is derived for a perfect Hexakis Icosahedron. For irregular shapes, different methods may be required.

Q5: What units are used in the calculator?
A: The calculator uses cubic meters for volume and square meters for surface area. Ensure consistent units for accurate results.