Total Surface Area Of Hexakis Icosahedron Given Medium Edge Calculator

Formula Used:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \left( \frac{22 \times l_{medium}}{3 \times (4 + \sqrt{5})} \right)^2 \]

Total Surface Area of Hexakis Icosahedron:

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

The Total Surface Area of Hexakis Icosahedron is the amount or quantity of two dimensional space covered by the surface of Hexakis Icosahedron. It is an important geometric measurement for this particular polyhedron shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{44} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})} \times \left( \frac{22 \times l_{medium}}{3 \times (4 + \sqrt{5})} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Hexakis Icosahedron (m²)
\( l_{medium} \) — Medium Edge of Hexakis Icosahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the total surface area based on the medium edge length of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this shape.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, structural analysis, and geometric studies of polyhedral shapes in mathematics and engineering.

4. Using the Calculator

Tips: Enter the medium edge length of the Hexakis Icosahedron in meters. The value must be positive and valid for accurate calculation.

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, featuring 120 faces, 180 edges, and 62 vertices.

Q2: What units should I use for the medium edge length?
A: The calculator uses meters as the default unit, but you can use any consistent unit as long as you maintain consistency throughout your calculations.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the Hexakis Icosahedron, assuming exact input values.

Q4: Can this formula be used for other polyhedral shapes?
A: No, this specific formula is designed only for calculating the total surface area of a Hexakis Icosahedron given its medium edge length.

Q5: What if I have the short or long edge instead of the medium edge?
A: Different formulas are required for calculating surface area from short or long edges. This calculator specifically uses the medium edge length.