Total Surface Area of Hexakis Octahedron given Midsphere Radius Calculator

Formula Used:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{4 \times r_m}{1 + 2\sqrt{2}} \right)^2 \]

Total Surface Area of Hexakis Octahedron:

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

The Total Surface Area of Hexakis Octahedron is the amount or quantity of two dimensional space covered on the surface of Hexakis Octahedron. It represents the sum of the areas of all the faces of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{3}{7} \times \sqrt{543 + 176\sqrt{2}} \times \left( \frac{4 \times r_m}{1 + 2\sqrt{2}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( r_m \) — Midsphere Radius of Hexakis Octahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the midsphere radius, incorporating mathematical constants and geometric relationships specific to the Hexakis Octahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is fundamental in various fields including mathematics, engineering, architecture, and materials science. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the total surface area using the specialized formula for Hexakis Octahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cube. It has 48 faces, 72 edges, and 26 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is defined as the radius of the sphere for which all the edges of the Hexakis Octahedron become a tangent line on that sphere.

Q3: What are the practical applications of this calculation?
A: This calculation is used in geometric modeling, architectural design, crystallography, and in the study of polyhedral structures.

Q4: How accurate is this formula?
A: The formula is mathematically exact for a perfect Hexakis Octahedron shape and provides precise surface area calculations.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the standard unit. For other units, convert your measurement to meters before calculation.